Homeostasis is a giant catch-all term for normalizing things in or about the body. In this context, something foreign is introduced to the body (thorn or the tattoo ink) which is affecting the “normal”(equalibrium) state of the body. The body will then do its best to return to this equilibrium, and in these cases, that involves expelling (thorn) or slowly removing (dye) the objects from the body.
I’m going off of memory, but homeostasis also covers our body temperature and chemicals. It’s why medical personnel can take blood and learn about issues; there is an expected range for everything to be in. Homeostasis is just that over-all term for “things should be this way”. There are dozens of equalizing processes under the term “homeostasis”.
I’m generally a fan Swift’s music, but I remember listening to this song and just thinking it was the feminine version of a white night incel. I get it’s supposed to be the song “all of us” felt like when we were in middle school or high school watching a crush date someone, but anyone older should view the song as a “what not to do” song.