If a Jew owns an animal, are they permitted to give that animal a tattoo? I would imagine this is permitted, but are there any opinions it is not allowed on the principle that a Jew shouldn’t do something to others that they are prohibited themselves?
tl;dr it is probably not allowed to tattoo an animal, but only because of the mitzva of not being cruel to animals, but nothing to do with the lav of etching the skin (tattooing). CYLOR.
According to Sefer HaChinuch, the purpose of this mitzva (following Rambam) is to distance ourselves from idolatrous practices. The reason why idolators would do this practice is an act of submission to their idol, as if they are marked for its service. This root of the mitzva is non-applicable to animals.
Rambam explains this applies to men and women (but not animals). The Shulchan Aruch YD 180:4 explains that it is technically allowed to tattoo one's servant, all the moreso one's animal.
You ask if it is prohibited because of a potential prohibition of not doing to animals what we wouldn't do to ourselves, but there is no such prohibition. There is a prohibition of "Tzaar Ba'alei Chaim", which is to not cause unnecessary pain to an animal, which could very plausibily be invoked by a posek if a person asks if they can get a cosmetic tattoo on an animal, because it can be considered unneccessary. Branding might be the only practical exception that would be allowed that fits with the question, although I'd imagine some poksim would follow the Rema (SA EH 5:14) and encourage the farmer to try and find a less painful method to identify his animals, such as with paint or an ear-stud label.
"You shall not make cuts in your flesh for a person [who died]. You shall not etch a tattoo on yourselves. I am Hashem."
The verse prohibits us from tattooing ourselves, and it does not prohibit tattooing our animals.
However there may be a different issue involved with needlessly tattooing animals, as it might expose them to unnecessary pain, which is prohibited due to "Tzaar Ba'alei Chaim".