Question: Is there a “me too” fallacy?
This can also be avoiding the issue (see https://www.logicallyfallacious.com/tools/lp/Bo/LogicalFallacies/61/Avoiding-the-Issue). The issue brought up is not addressed, rather a new issue is raised. A good response might go like this.
Me: It hurts me when you do this...
Them: Well it hurts me when you do this..
Me: Let's address that, but first, lets address my issue then we will get to yours. Sound fair?