An LM386 indeed is an op amp, but it interfaces external components for feedback in a different way, making it non-compatible with most standard op amp applications. The LM386 includes some of the feedback network of its own internally, and the external components merely complete the feedback network so that the designer can modify the gain.
In short, the diode compression op amp could be used in something like a Little Gem. It would require the addition of input DC biasing, and a higher power rated output section if it's going to be compatible with the Little Gem circuit. It is not a project for a noobie since it requires some electronics knowledge to redesign the circuit to work in a little gem.
Again, this circuit is a good replacement for op amps in a TS, or a Dist+, etc. It requires all the input biasing and DC decoupling treatment that any audio electronics op amp application would require. The results from using this as an effect would be about as disappointing as connecting a power supply to an IC op amp, your guitar to one input, and the amp to the output...sputter, fuzz, pop...or maybe nothing.