Right now the common tests have two different use-cases: checking internal sklearn estimators, and checking 3rd party estimators via check_estimator.

The common tests are very strict to a degree that's not really reasonable for 3rd party estimators, in particular in checking for particular error messages.
Furthermore, the required error-messages are not documented anywhere (see also #10082).

I think it would be good to add a strict=False argument to check_estimator to not require exact messages, and possibly not even the exact error types. That would make it much easier for people to use check_estimator.