Conceptually, it would be akin to having say, France sue Italy, even though both are in the EU.
Say for instance, Italy selling bubbly wine and calling it Champaign.
Well, that's already covered. Not the country of Italy, itself, but the producer/vendor falsely claiming a 'champagne' outwith the area of Champagne. And then usually challenged by the local equivalent of Trading Standards (or Customs, if imported) in any country that cares to enforce the PDO/PGI/whatever. (And given that Italy has its own Protected origin stuff, it's unlikely not to clamp down on its own bogus producers if it finds out first.)
The ECJ is probably the 'SCOTEU' we'd be talking about in equivalent state-vs-state legal battles (though it's usually individiduals or effective-individuals sueing for a overturning of a (different) state's actions or inactions they claim as 'unconstitutional' or similar). Despite what many say, inter-EU disagreements at state level
tend to be done as normal national diplomatic business between the sovereign states, not utterly controlled by the apparently all-consuming 'bEUrocracy' of shape-shifting lizards.
The closest equivalent I can think of, off the top of my head, is the EU parliament officially decrying (ineffectially?) the recent Polish internal election process/outcome, but that falls short in several important ways.