Mick's comment is on the right track here. I've experienced similar things, with cats that were arguably even worse off; so I hope my experiences can help you in approaching your problem.
For reference:
We have two cats who were born in the wild, and lived there until they were 6-7 months old. They lived inside a prison building, which means that they grew up needing to avoid human contact at all costs.
They were taken into foster care (a home specialized in teaching cats what it means to live with humans), then another foster home (looking for adoption), then we adopted them.
Obviously, they were incredibly shy when they first arrived. They snuck around, making no noise, and refused to move (or even blink) when we were anywhere near their room.
Having lived as feral cats inside a prison means that avoiding humans at all costs is ingrained in their behavior. From what we were told, siblings of theirs were caught at an earlier stage, and never to be seen from again (from the cats' point of view). This made it very clear to them that humans are dangerous.
They've been with us for 7 months now. Although we're somewhat convinced that they'll always remain shy cats, they have at least accepted us as their family, which means their fear of humans in general has been overcome.
We can give them pets, they sometimes come for affection (but not as often as the average cat), they follow us around the house, ... But we're not allowed to pick them up, we haven't gained that level of trust yet.
What we did:
The core intention of bonding with them is simple. You want to teach them that bonding with you has certain advantages. If they want the advantage (e.g. treats), then they must pay the price (socializing with you).
Because of this, one of the first things you need to do is limit their freedom. This sounds harsh, but you should keep some freedoms as rewards for bonding with you.
If you already allowing the cat to eat and otherwise ignore you, then the cat has no reason to ever want to bond with you.
Never ever deny them essentials (food, shelter, litter box). The only incentives you can use are bonuses (extra food, treats, toys). If you're using essentials to get them to interact with you, then they aren't learning anything about you, because they are purely driven by their survival instinct.
You need to bond with them over bonuses, because they choose to enjoy the bonuses. Their freedom to choose is quintessential to the bonding process. They must choose to interact with you, while having the option to not interact with you. If you blackmail them by denying them essentials, then they don't have the option of not interacting with you, and it will detract from the bond you want to have with them.
Over time, they will understand that socializing with you isn't all that bad (if they have no bad experiences, of course!), and they even get bonuses on top of that; so they have little to complain about.
Territory:
- Our house has many nooks and crannies. If we had set them free from the beginning, they would've hid and never come out. Instead, we gave them a bedroom all to themselves (with lots of toys and sleeping spots).
- Further increases to their territory would be awarded at certain milestones.
- After they were at ease with us in the room, we let them explore the hallway/staircase.
- After they let us pet them during food/treat time, they were welcome in the bedroom.
- After they had bonded with us sufficiently, we gave them access to the entire house.
- We currently don't allow them out, but are likely to try it once they've been with us for a year.
Food:
- The first few days, we put food in their room, closed the door, and let them eat in peace. They needed to adjust to their surroundings.
- Afterwards, we stayed in the room and observed them from a distance. There was at least 3m between us and the food. They only ate while keeping an eye on us, and would scurry for the slightest move we made.
- Then, we gradually decreased the distance between us and the food, half a meter every day. We were in arm's reach within a week.
- One of the girls was less eager to come close, compared to her sister. So we split her food over two bowls. One bowl (with a little bit of food) was put at a distance, but the other bowl (with the majority of food) was put next to her sister. She got to choose: shy away from us, or come and get the rest of her food. Within three or four days, she opted for getting the rest of her food.
- Then it became time for them to get used to petting. Whenever they would eat, I would carefully approach them (in a way that they saw my hand approach them) and gently stroke the back of their neck. They obviously refused that in the beginning.
- Whenever they recoiled, I pulled back my hand and looked away, so they could eat in peace again. I let them eat some of the food, and then tried again. I spaced it out in a way that I could try it about 5 times by the time they finished the food.
- Eventually, they realized that we weren't hurting them in any way, and the inconvenience of recoiling and no longer eating outweighed the inconvenience of being pet by us. Therefore, they started allowing us to pet them as long as they were eating.
- When the food was gone, so were they. They didn't like being pet, but they at least allowed it when they had bigger fish to fry (= eating).
Treats
- Treats are extras. While we allowed the cats to initially eat in peace, since they need the food, we immediately put a price on the treats. They don't need the treats to survive, but they want the treats because they're tasty. This creates the perfect incentive for them to want to conquer their own fears, without risking them getting malnourished.
- The first treat was given for free. But they needed to allow me to pet their head once if they wanted another treat. After they ate their treat, I slowly attempted to pet them. If they refused, I let my hand hang in mid-air, continually offering it (without approaching them again). It didn't take too long for them to figure out that letting us pet them gave them another treat.
- They didn't understand the "cost of petting" initially. So I put a timer on it. Without petting, they would get a treat every 30 seconds (give or take). But if they allowed me to pet them once, the timer would immediately reset and they'd get a treat. Since our visits were somewhat fixed length (due to other obligations), they quickly learned that petting means getting more treats during a visit.
- One of them was a bit slower than the other. She did come forward and sniff my hand, but wasn't at the point of petting yet. So I allowed her to have another treat if she sniffed my hand for a while. She soon figured out that petting yielded treats faster than sniffing my hand, and she switched her approach.
Toys and playing
- Initially, they got a handful of toys (play mice, balls).
- We introduced new toys. Whenever we appeared, we would bring a few toys with us. But we also took them back with us when we left.
- Try to put a limit on the playtime. If you're available for playing at any time of day, they won't always come forward quickly. But if you limit playtime to one or two scheduled visits (e.g. morning and evening), they will have to make sure to play during those limited visits, which increases the need for interacting with you (and thus being happier to see your presence);
- Though we started leaving some toys behind; we intentionally kept the good toys (the ones they clearly preferred) with us; so that they connect the happiness of playing with that toy to our presence.
- A good tip is to put the toys in your pocket before you enter their room. They will quickly learn that you pull them out of your pocket. Their eagerness to play with their favorite toy may override their tendency to keep you at a distance; which is what happened for one of them.
- Besides toys that they can play with by themselves, try to get them hooked on toys that require your interaction (laser pointer, fishing rods, ...). Ours loved the laser pointer, and quickly identified that they needed us for that.
- Once in a while, stop playing with them sooner than usual. Try and see if they approach you as a way to ask you to continue playing.
Random tips
- In the beginning, you'll want to put the cat in a familiar environment, where the only alien object is you.
- After a while, try to introduce objects that are even more foreign to them; which makes you "familiar" (comparatively speaking). For our cats, showing them a tablet that was playing cat videos was so interesting to them, that they almost forgot about us. They absent-mindedly would come closer to us than usual, because they were focusing on the tablet.
- Tailor your approach to the cat. I listed things that worked in my experience, but your cat may have completely different likes/dislikes. See what they like, and try to manipulate the situation so that they have to bond with you in order to get access to what they like. Really, it's as simple as a "carrot and stick" approach.
- This will not be fixed in a matter of days. This is a very slow progression of trust building. If you take things slow in the beginning, you'll eventually see them increase the rate at which they accept new things. That's a prime indicator for you to see that they are actually bonding with you. The fact that they accept a new thing faster, means that they trust you to not endanger them.
- It's possible that the cat simply doesn't like socializing. Though this can be a sad realization as a pet owner, you should also understand that the cat is not to blame for it. Often, you can eventually get them to come around to it, but that's not a guarantee.
In general
The general approach is always the same:
- Guarantee the basics. They don't need to do anything they don't want to do, you will provide for their basic needs.
- Present two options. The option where they avoid you and do not get a bonus; and the option where they interact with you and get a bonus.
- Let them choose. If you force their hand, they didn't make a choice, and will not learn anything from it.
- Reward progress. If they reach a certain milestone, give them an even bigger reward. Note that if they consistently achieve the same milestone, they don't get big rewards for it anymore. Don't give bigger rewards for stagnating behavior, as it teaches them the wrong thing.
- Do not punish lack of progress. This makes you into the boogeyman. To them, you are being unkind to them for no reason (since they don't know what your expectations are).
- Accept regressive behavior. If they're suddenly more shy than the day before; so be it. Cats have different moods, just like humans. If they don't want to play today, that's their choice.
- Don't break the rules in their favor (or yours). If they don't want to play today, that's fine. But don't try again in a little while. Stick to your schedule. Make them understand that they make their own choices, but suffer the consequences of that choice.
This is everything I can think of right now.
