"...what makes one opamp swappable with another." That's the kind of things you ask engineers... But for most stomp box circuits:
Some have higher input impedence (MOSTET vs JFET vs BJT)
Some have "faster" "slew rate" ("slower" might make a "softer" distortion)
Some operate "rail to rail" or very close to the power and ground "rails." (ones that don't might either distort very unpleasantly or be useful for their "gating."
Some use less current, better for battery life or keeping an LFO from being noisy.
Things like offset null on some single pampas...small differences in DC output can build up over multiple stages and you can correct that.
You can read about opamps, somewhere you'll read something like "The perfect opamp has infinitely high inout impedance, infinitely low output impedance, open loop gain of infinity" and other stuff I've forgotten. They're very good, but never perfect.