It comes from the world of systems operations. Something long-lived and trusted, so high emotional attachment (pet), vs. something short-lived that thus does not need to be trusted, so comparatively low emotional attachment (cattle).

For example, Bob's one-of-a-kind trusty server from which Bob is nigh inseparable, vs. a Docker container with a version controlled config you routinely tear down and bring up instances of, maybe even in an automated fashion.

Here this would map to trusty aged codebases you don't touch out of fear and caution, vs. codebases you can confidently touch because the spec, the code, the tests, the tooling, and the processes are solid.