Neither you nor your instructor is entirely correct.
I would prefer to find an independent flight instructor to continue my training and to book a skills test independently.
That is not possible under current EU legislation. All training for the issue of a licence must be done within an ATO and the ATO providing the training must recommend you for the theoretical knowledge examinations and the skill test.
I have been told by my flight instructor that I am legally obliged to complete all training for a PPL licence with the same institution I began with.
That is complete nonsense - your instructor is either an idiot or a con-artist! There is nothing to stop you moving to another ATO, although, depending on the terms of your contract, you might not get the balance of your ill-advised, up-front payment back.
Oh, and did somebody mention -
NEVER, EVER, PAY UP FRONT!