It sounds like there are a couple of issues here. One (and we really don't have the information here) is whether the friend who got the tattoo presented it as something like "hey, look, I got your symbol, it connects us" or whether it's more like "I've always loved Kermit the Frog, and seeing you guys drawing him on stuff prompted me to get this tattoo." ("A character" might be a muppet as easily as a monogram.)
The LW gets to figure out whether she still feels comfortable with this person, and maybe even to say "I know you meant it well, but it feels like you're trying to connect yourself as part of my marriage, which is weird. Your body is your business, but would you do me a favor and cover the tattoo when we're hanging out together." Or to just back away, if it seems like part of a pattern of trying to be too close.
But in the end, it's the other woman's body, and etiquette doesn't justify asking, let alone demanding, that a friend get a cover-up tattoo. (There are three people in my life who I have told that they can veto any tattoo I'm thinking of, if the specific image bothers them. But that's not something the rest of the world gets.)