Not necessarily. In some places the animal abuse laws are rather specific. E.g. in the infamous Enumclaw case (Washington, U.S.) they failed to prosecute for bestiality because there was no law against it:
https://en.wikipedia.org/wiki/Enumclaw_horse_sex_case. The most they got away with "trespassing". Of course, they passed a law banning bestiality quickly following the press coverage of the story.
There are other examples of cases where this has been "tested" in court, and the court was forced to let the accused person go. I think up until recently in Germany bestiality was clearly legal based on precedent set by a previous ruling? I know for a fact that bestiality was legal in Denmark until 2015.
This I very much agree with. I'm mainly interested for curiosity's sake. If OP is planning on moving somewhere that doesn't have a law against bestiality, I would strongly advise to reconsider, because the law can change at any time (and in many cases it's unclear whether or not it's
illegal as you described).