The common case for catching ImportError is exactly what ModuleNotFoundError represents.  If people are already going to change their code to handle just this special case, I'd think having the subclass would be the better route.  I find it simpler (both to update existing code and to read:

    try:
        from _thread import _local as local
    except ModuleNotFoundError:
        from _threading_local import local

vs.

    try:
        from _thread import _local as local
    except ImportError as e:
        if e.not_found:
            from _threading_local import local
        else:
            raise

And for the broader case:

    try:
        import breakfast
    except ModuleNotFoundError:
        class breakfast:
            spam = 0
            eggs = 1
            ham = 2
    except ImportError as e:
        log_some_message("uh oh: {}".format(e))
        raise

vs.

    try:
        import breakfast
    except ImportError as e:
        if e.not_found:
            class breakfast:
                spam = 0
                eggs = 1
                ham = 2
        else:
            log_some_message("uh oh: {}".format(e))
            raise