Is the Basel/Mulhouse airport in Switzerland or France?
Yes, both. One half of the terminal is in France, one half in Switzerland. You can cross between the two countries inside the terminal at departures level, but not by road. The Swiss half of the terminal (technically leased from France) is connected to Basel city via a road that is separated from other roads until you enter Basel City limits and are in Switzerland proper.
If coming from Colmar by train you would get off at St Louis station, and take a bus to the French side of the airport.
If you rent a car from the Swiss sector (cross from France inside the terminal), you get a Swiss registered car at Swiss prices with a Vignette, and exit you on the road to Basel.
If you rent a car from the French sector, you get a French registered car, pay French prices and it may or may not have a Vignette, and you exit to the French Autoroute, two junctions before the border.
I think it is just easier to hire your car in Colmar, and buy the Vignette at the border.
Yes, Basel has a crazy relationship with borders ☺
P.S. The Vignette costs CHF 40 (NOT 50).