If you grip the inside of your lips between your teeth and allow them to move back through, you can feel many small, relatively tough structures within. What are these structures, as I compulsively bite at these to the point of removing them, and would like to know what they are in case biting them free is causing serious harm. Biting in through the outer surface of the inside of the lip reaches a fairly strong membrane/barrier, and biting into this barrier allows the aforementioned structure to slip through it into the mouth, where it can be bitten free.
On this diagram I would say it appears to be either the orbicularis orbis muscle, or minor salivary glands. The muscle's separation from the main flesh of the lip and membrane between the lip and affected area seem to match up, but if these muscles are fibres then it is unlikely to be those.