Usually, financially stable parents who pay for their kids’ clothes, phones, cars, etc. will say, “I paid for it, so it belongs to me.”

Financially “stable” just above the poverty line or formerly poor parents, yes.

In my experience, rich people don’t look at the bill or who owns what.

They will occasionally use the concept of having supported someone financially as a cudgel, depending on the person, but they will never go “well those shoes I bought for you are still mine because it was my money” cause the shoes don’t matter nor does the money used to buy them.

It is literally “the thought that counts”, because the dollar amounts are trivial.

As for who owns what in the broader sense of family fortunes, that really depends on what kind of wealth and the culture they are from. Are they old money? are they nobles? are they new money? This will largely dictate who is actually in control of the “family” holdings.