As a general rule, it is unwise to ever give a pet as an unsolicited gift.
A pet is a large commitment, and there are animal shelters full of unwanted Christmas, Easter and birthday gifts.
A puppy, kitten, bunny; any pet is a living being and totally dependent upon their owners in all ways, they are a lifetime commitment. Too often, life's circumstances will force dedicated pet owners into the difficult corner of having to surrender their loved pets.
Deciding and choosing a pet is a decision which takes time and forethought on the part of the would-be owner. Facilities, lifestyle, time, and financial constraints are unique to that household; an outsider can only guess at these factors.
Families may have many varying restrictions on whether or not a particular, if any, type of pet is suitable for that family, so an outsider buying a pet for a family or person, does not constitute responsible gift giving.
As for parents buying their children pets as gifts, any wise parent knows that the responsibility of a pet ultimately lies with the parent, just as the responsibility of the child is with the parent. So if a parent chooses to give their child a pet as a gift, at any time of the year, the parent needs to know and accept responsibility for that pets well being.
So, to decide if you are gifting a pet, you need to replace these question:
1) Is this pet a suitable gift for [insert person]?
With this more important question:
2) Am I going to be caring for this pet?
If the answer to the second question is "No", then the answer to the first question is also "No".
Many pets are given as baby pets while they are cute, young and healthy, a pet remains dependant upon their owners as they grow old, sick and have behavior that does not always suit our lifestyles. Maintain an animal within a human household requires effort. A pet doesn't attribute value to material things as people do, so when a cat is scratching up furniture, or a dog chews an expensive pair of shoes, or a bunny leaves dropping all over the house, this all requires effort, patience and expense, that can make the reality of owning a pet outweigh the cuteness of the original gift.
Unless you have prearranged with someone , with whom after long consideration and consultation, has agreed and desires a particular pet, a responsible adult who is prepared to make the lifetime commitment of caring for and homing the pet and together you have chosen this pet; the answer is always a categorical no pet is suitable to give as a gift.
My personal examples:
1) My neighbors took my daughter for a walk (as they often do), but this time returned with two goldfish in a small goldfish bowl.
I was dubious about the potential survival of the fish, as the bowl was small and we had nowhere to place the bowl where it would not be bumped.
After the first fish died, we had our neighbor add the surviving fish to their pond.
The moral of the story: No matter how easy caring for a pet may seem, do not give pets as unsolicited gifts.
2) I was given a bird, a budgerigar as a child. The bird was I failed to feed and provide the bird with fresh water and he died. My mother then used the event as a let this be a lesson to you, birds need fresh food and water daily. I then placed food and water beside the dead bird (it didn't help). This I have never forgotten, and although my children have pets, and I prompt and remind them to feed them, the responsibility lays firmly at my feet to ensure that our pets have daily fresh water and food.
The moral of the story: No matter how responsible or ready our children may seem to be for pet ownership, the responsibility, ultimately, lies with the parent, so gifting a pet to a child, in reality means gifting the pet to the parent.