Based on the content of the question I assume it is more about the legal definition of ownership than about a "universal" definition of ownership based on some form of natural law.
The idea that the law as upholded by a government defines what ownership really is is called Legal Positivism and is opposed to the theory of Natural Law or Jusnaturalism, the idea that there is a definition of justice independant of the government and that laws only try to approximate this ideal justice (be it God given, dependant on Human Nature, or Reason, etc...).
If we go with legal positivism, property rights are in fact not monolithic but divided in several categories to handle a bunch of specific cases that couldn't be settled with a single general concept of ownership (just like the mortgage case in OP's question). The division below takes root in ancient Rome's legal system, which became the base for property laws in most of the western world and, from there, influenced most of the world.
Property is split in 3 main categories, Usus, Fructus and Abusus.
Usus is the right to use a thing. Live in a home, drive a car, use a tool to do some work. This one is pretty straightforward.
Fructus is the right to own what is produced with what you own (the fruit of one's work with this thing). For example, the yield of a crop field, the benefits of a shop, the offspring and milk of a cow, the dress you just made using a sewing machine, etc...
Abusus is the right to modify or destroy a thing. Demolish a house, or build one in a certain land, slaughter a cow, dispose of a car, etc...
Those rights are combined in a variety of ways to cover for the nigh infinite diversity of commercial deals that can be made of goods.
For example, I could be a landlord owning a field (in the colloquial sense, having "ownership" of the field). A common form of deal is to lend this field to a farmer in exchange for a yearly rent, under the condition that everything that grows on the field is theirs. By paying the rent I grant them the usus (they can enter the field to work it), and the fructus (the crops is theirs), but not the abusus (of course they can't use the field to build a condominium or something, they have to give it back as is). Yet I also forfeit part of my abusus (I can't just decide to build a condominium either, or I need to compensate the farmer for the loss of their crops).
The farmer could even have an employee, with whom they share part of the usus (the employee can work the field), but nor the fructus (the crops belong to the farmer), nor the abusus (the farmer does not have it in the first place).
All the details of this depend on the actual legislation at the time and place of the deal, but those are the general principles. We can see that it is not clear cut who has ownership of the field. I think in layman terms most people would agree the landlord keeps ownership because at the end of the deal, if everything go as planned, they will recover all 3 types of right.
How can we apply those principle to OP's mortgage exemple?
Usus is definitely detained by the buyer. The buyer leaves in the house, and the banker who owns the mortgage can't enter the property uninvited les they'd be commiting trespassing.
Fructus in the case of a residential house is hard to define, but if we assume it is a shop then all benefits would also go to the buyer. The mortgage owner can't claim any of the shop's benefits outside of the payback that was agreed upon.
Abusus is less clear cut. If the buyer was to significantly alter the value of the house (say, by destroying it), the mortgage owner would defnitely suffer some financial damage. I am no lawyer and no juridiction is specified in the question, but it seems obvious that nobody would ever give the buyer a mortgage if they can just burn the house to the ground. So the right of Abusus is probably shared between the buyer and the bank, or limited in some way by the mortgage contract.
We can see that the buyer enjoys most of the property rights linked to the possession of a house. Also at the end of the deal, if everything goes as planned, the buyer will have full property of the house.
Note that in other specific cases, like an appartement in a condominium, the buyer will never have full ownership rights over what they bought (you can't just destroy one appartment of a building without compromising the property of all other appartment owners). Even in the case of an individual house, they might be forever subject to Neighboor Association rullings like not being allowed to have a gazeebo or barbecue on their front yard, etc... (it is to say, their usus will forever be limited).
Wether it's appropriate to say the buyer has "ownership" of the house despite not having full property rights is left for anyone to decide.