From the rules for Monopoly (emphasis mine):
THE BANK… Besides the Bank’s money, the Bank holds the Title Deed
cards and houses and hotels prior to purchase and use by the players.
The Bank pays salaries and bonuses. It sells and auctions properties
and hands out their proper Title Deed cards;
BUYING PROPERTY… Whenever you land on an unowned property
you may buy that property from the Bank at its printed price. You
receive the Title Deed card showing ownership; place it face up in
front of you.
This is how the ownership of properties and title deeds work, the holder of the card owns the property. At the start of each game, all properties are unowned, the bank has all the cards. As the game progresses, each player is given the option to buy the unowned properties they land on, if they do the bank gives them the deed to that property and they now hold the deed, showing they own the property.
As for the paying of rent, you don't need to remember anything about the owner of the property for paying rent, it's not up to you to just pay rent in Monopoly, it's up to the owner of the property to ask for it:
PAYING RENT… When you land on property owned by another
player, the owner collects rent from you in accordance with the list
printed on its Title Deed card.
The owner may not collect the rent if he/she fails to ask for it before
the second player following throws the dice.