Would it be safe to use the water I take from my dehumidifier and give it to my pets?
No. The typical collection tank from a residential dehumidifier is not a safe source of potable water, and certainly isn't up to the quality of commercially-distilled water.
Examine Your Options
Water from a typical dehumidifier is not "distilled." It is extracted through either condensation or dessication of ambient humidity. In the latter case, the extraction process itself will generally make the water non-potable; in the former, bacteria and mildew can build up in the collection tank. Every residential unit I've personally owned has carried warnings that the water is not potable.
Even without the warnings, I wouldn't use dehumidifier water directly. If you want to re-use the water for some reason, I'd pass it through a solar still, solar water purifier (e.g. a Puralytics SolarBag), or LifeStraw first, and make sure you scrub out your dehumidifier tank regularly with soap and water to limit the build-up of mildew and sludge.
A cheaper and safer option might be to install an on-tap or in-home water filter that removes chlorine and flouride from the water supply, and can provide whatever additional level of purity you need. However, if you truly need the purity provided by distillation and/or reverse osmosis, then you'll have to do some price shopping to see if doing your own filtration will be cost-effective for you.