proof/certificate of ownership of the property

I would think that this is a question for the notaris who acted for you - you paid them enough!
In any case, if there is an outstanding mortgage, you don't own your property in full.
Also, I have no memory of getting such a document when we paid our mortgage off but it was some years ago so my memory might be faulty.

Your notaire.

If you recently signed an agreement of purchase you will eventually have to sign the "Acte". This is a document which describes the history of the property. At the time of the signing of the "Acte" you will get a copy of this document from your notaire. This is the proof that you own the property. The signing of the Acte has to happen within four months of the signing of the purchase agreement. If you have already paid all of the money ask your notaire.

We got such a certificate when we signed the final act for the purchase of our house.