There are certainly some double standards going on here.
I get it Markor - it's those little snarks that individually mean nothing, but over 20 years build up to a whole heap of irritation. And I wouldn't accept a "Yeah whatever" from anyone, it's dismissive and rude. People react to irritations in different ways - some don't find them irritating at all, others do. Both are fine - we are allowed to feel how we feel, so although others have dismissed your irritation, I accept it's what you feel and that's OK.
For what it's worth I do refer to houses and bedrooms by the couples' names, and sometimes for brevity, by their joint surname if they share one. So 'We're off to Joan and Jim's this weekend' or 'the Smith's are having a new kitchen'. It's certainly just as quick to say "The dog's in their bedroom" as the "The dog's in Joan's bedroom".
One thing I would say - your wife needs to pick up some slack here - it's not good enough to say "You know whats she's like". I know she's spoken to her DM about it, but she probably said to her "You know what he's like".
How to manage the situation going forward. This has to be in conjunction with your wife. Tell her her DM's snarks, although individually small, mount up and are now at the point where they are starting to adversely effect your relationship with your MIL. Ask her to help reset the relationship, eg meet on neutral ground, no staying overnight, stay at PIL's (and resist the tempation to assign ownership to just FIL!). Ask if she can explain your feelings to her DM in this way and to resist the "That's the way she / he is" reason, as it's not a reason, it's a cop out. At the end of the day, you can't change her or her behaviour, only how you react, so if a reset doen't work, you could step away, stop helping out so much, etc