The Colt is only used during this overhead shot, and remains the S&W Model 36 when he is shooting.
Most casual daters report to find their match/es online, surprisingly not in a bar.
Bickle buys it after "Easy Andy" suggests it as a more practical weapon. A mobster in the whorehouse during the final shootout has a blued Model 36, which Travis takes from him and uses.
In a continuity error, the Model 36 becomes a nickel Colt Detective Special when Travis puts it down in front of him at the range (see below). The pistols that lay in front of him are the S&W Escort (can be told apart from the Sterling by the bottom barrel) and the S&W Model 36 (can be told apart from the Colt by the 5 round cylinder). Note the barrel, ejector rod, and six-round cylinder of the Detective Special.
They said this measure increases the chance of compatibility.
There is also the more established Adult Friend Finder, which is explicitly for those seeking out like-minded sexual partners.