We had a mortgage for 30 years before paying it off recently on our modest home. Now we consider ourselves home owners.
Dd rents because she doesn't want to pay the bank more in interest than the cost of her rent is. Yes her rent can go up but so can your mortgage.
I've had friends of mine make snide comments that dd is going on another holiday when she doesn't own her own home. This is as opposed to their own children who have recently bought with 95% or 90% mortgages in a part of the country where a 3 bed house is less than £150k.
AIBU to think that you don't own your own home just because you have a £15k down payment. DD has far beyond what is needed for a deposit but it makes zero sense in central London at present.