There is no limit on how many houses you can own in the Elder Scrolls Online, basically you could buy all the houses that are available in the game if you have the funds. Some of the houses are crown store exclusive though, that means if you truly want to get them all you will have to pay real money for it.
There are two ways to obtain houses, either via ingame with gold or via the crown store with crowns aka real money.
Ingame Aquisition (Gold)
There are also plenty of houses available ingame that you can buy with gold that you during your journey in the Elder Scrolls Online. They usually just come unfurnished and depending on the size they can cost quite a bit of gold, the bigger ones can cost several millions.
Crown Store Aquisition (Crowns)
If you purchase houses in the crown store, you can either get them already furnished or empty. If you buy them furnished, you will pay extra crowns! I would recommend buying houses unfurnished, because you will most likely decorate your house yourself anyway.
If you want to see which houses you do own, and which ones you still need to get, you can go to the collections section ingame (default button U) and check all the houses. You will have to navigate to the "Housing" section on the top right side.