There may be some exceptions. For example, at New York JFK, some airlines have their own terminals.
they dont build them, they simply occupy all of the slots in them.
In JFK there's only a single airline that has "it's own terminal", and that is Delta @ Terminal 3, every other terminal has multiple occupants (Terminal 8 seems to be almost entirely AA but I believe I've seen other oneworld partners there, regardless they didn't pay its recent reconstruction). And they actually do own it, but they didn't build it, having bought it as a result of Pan Am's bankruptcy. And it's actually being replaced starting this year with a new Terminal and the old one is being demolished. Delta is not
paying for the new one outside of a very
small percentage (low single-digits) of the cost. News reports almost always say a terminal is being built "by" an airline, but that's virtually never the case -- reality is that the port/transit authority operating the airport is putting up 90%+ of the cost via bond issuances, most of the remainder is being paid by facility usage fees, and some tiny
remainder is paid for by airlines.
Bottom Lines: The practice of owning ones own terminals is exceedingly rare and terminals have virtually nothing to do with slots anyways except the rare cases of flights backing onto runways because there's no open gates. Slots are a function of runway capacity and airlines do not
build runways. They might campaign for them, but they do not build them.