Mortgage interest is payment for renting the money, so it can be considered a substitute for rent payment on rental housing. Property owners also have to pay property tax, which is somewhat embedded in the rent payment on rental housing (of course, in California, Proposition 13 is like the property owner’s analogue to rent control).
Homeowner insurance replaces renter insurance, but also includes insuring the building (that cost is somewhat embedded in the rent payment for rental housing). Maintenance on an owned house is out of pocket for the homeowner, but embedded in the rent payment for rental housing.
The non-analogous financial aspects are that buying a house means making a highly leveraged investment in real estate, which may result in big gains or big losses, and that the transaction costs relating to buying and selling a house are much greater than moving to different rental housing.