They're not obligated per se, but a good one will refuse to work on you if you get too many too fast; part of this is a professional consideration--each tattoo is effectively an open wound and creates stress on your immune system, meaning you're at risk for slower healing and infections. A lot of (good) artists don't want that, as any sort of damage to the tattoo will affect the appearance and therefore their "advertising".
Tattoos give you a really great feeling of satisfaction; if you're someone who recently got inked, it's common to catch "the bug" where you really want more tattoos and you are planning a (different) new one almost constantly. Smart people realize this stems partly from the massive endorphine release, and don't get another 2 immediately.
Anisa's hand tats would have just healed fully. She's definitely riding that post-tat high and chasing the inky dragon. Under normal circumstances, her artist might say something, but she's chosen LA money sharks with mid skills, not artists, and they dgaf as long as they get paid.