Yes. You keep any and all trainings you have regardless of the bio. But you can only use the training if you have the bio for that element, or skill.
For example, if you trained in C-rank fire and you drop the bio, once you get a new bio you get to keep that "C-rank fire" training and continue from where you left off, but you must have a bio to continue training.
However, if you trained in Sharingan with an Uchiha bio, and you drop that bio, you can't have Sharingan/Sharingan training and techniques in another bio that isn't an Uchiha.
Hopefully that wasn't too complicated!