"Keep on your studying", which you say "means you never give up and persevere"
You wonder what the difference is between "Keep on your studying" and "Continue learning" or which of the two phases is more spoken English.
First of all, "keep on your studying" is NOT quite correct English. I ' m sure this surprises you, * keep on your studying* should be revised to:
"Keep on studying"(no "your") OR
"Keep studying" OR
"Keep UP your STUDIES " OR
"Keep/ continue learning"
I don't really find much difference between "keep studying" and "Keep (Or continue) learning". As to which one is more common in spoken English , Probably the shorter word,"keep" (remember that spoken English usually prefers shorter words)
But please note that in English there is a considerable difference between "study" and "learn". In English these two actions usually follow the sequence of first, study, and if you study hard, then you learn. So, "Keep studying" means to keep being a student or keep TRYING to learn, whereas "keep learning" means to go on adding to your storehouse of knowledge, whether by being a student (i.e. by studying something )or by learning from your experience