My only piercing is a nipple piercing, which is a bar with a loop beneath on which I have a small metal tag saying "pig proper of my owner" and a number as though I'm cattle he has labelled (I mean, I am lol). It can be nice as a safe way to have a permanent ownership tag/label thats pretty subtle and hidden. We have a more typical livestock looking cattle tag which we have clipped onto my piercing in the past during sessions, which puts me into an extremely deep piggy mindspace. I have been debating a septum piercing loop, and getting around to getting my ears pierced and maybe clipping that tag to the ear. Unfortunately I have fairly conservative parents, so the septum piercing could cause drama. I have looked myself into getting chastity piercings, but unfortunately I have an extremely small clit hood, and it wouldn't be possible to get the required piercings for a clit shield for me, and while labia piercings are hot, we discussed and would miss possibilities with cunt impact. Not to say it wouldn't be possible, but we both feel it would be safer not to have piercings there, and also slapping metal would probably not feel good. Its always sensible to think about potential drawbacks of the body modifications, rather than simply allowing the horny brain to go for it lol.
A clit/hood piercing where a leash could be attached to would be a fun possibility. I know of a few subs who got some tongue piercings to augment oral sex, which could be of interest, but I have less to add on the topics of those.
In terms of tattoos, there is the possibility to do something called a bloodline tattoo (tattoo gun, minus ink). I had it done fairly large on my upper back and it healed/faded roughly 6 months later. We actually negotiated that I wouldn't know exactly what he was putting on me, but we spoke about direction of it before he did the design with the tattooist, discussing in a language I am not that fluent in, while I was sat in another room. It was hot and I had the fun of putting together what was written on me while we were at a kink club that evening (bonus thing, from how the skin raised and the font they used, it looked like a branding when it was fresh which was rather appropriate for the piggy vibe)
I know this was more asking for suggestions, but where I could add personal experience/concerns with my own explorations I thought i should add for some context.