The threads in the MTB frame that is supposed to hold the bottom bracket lock rings were very loose and it can't hold the bottom bracket in place. The previous bike owner tried to apply Teflon tape in it but the lock rings still moves out of the frame if pushed from the other side.
Any recommendations on what to do?
A video could really help, sounds like the threads of the cups and/or of the frame shell are damaged. Probably the Bottom Bracket cups need to be removed so the threads can be inspected for damage, like rust, or cross-threading, stripped threads etc. If the threads of the shell are in good enough condition, they can be cleaned and a new BB installed. If the threads of the shell are damaged, then it would be a bit more complicated and deserve it's own question/answer.
This image (from Park Tool website) has the names of some BB parts. Even if it's not a perfect match, editing the question and using these terms would make the question clearer.
I haven't seen one in quite awhile but Sunlite made a repair bottom bracket. It looked like a standard sealed square drive unit. The cups lacked the standard external threads. One of the cups was part of the bearing housing (shell). The other cup had female threads. The bearing shell had Male threads that screwed into the female threads of the cup. By using two bottom bracket sockets you tightened both sides to clamp it to the frame.
