Does a private party have the right to repossess a car for not doing the title transfer even though the loan was paid?

It depends upon the laws of the state in which the vehicle is sold and/or to be titled.

Engaging in a self help solution (repossession of vehicle) may not be the best choice, as it could possibly legally complicate matters further. The best option if for the seller to contact the state DMV explain the situation and obtain factual information according to the current laws.