Does cat removal disable check engine light?

You have misconceptions about how it all works together.

First, what @EᑎOT is accurate. The easiest way to deal with the ECU when the cat is removed is to put a dummy O2 sensor in place of a real one. If a "dummy" O2 sensor is in place of the original one (which actually works) it sends a constant signal back to the ECU in the "accepted range", which the ECU sees as normal, which won't set a CEL or throw a code.

Second, the downstream sensor which detects the function of the cat has nothing to do with how your car runs (NOTE: In newer cars, this isn't the case). It's just testing the efficiency of the cat (making sure it's working). If you remove the cat and "fix" it by changing the tune, the way it's done is by setting the parameters for which the ECU is reading from the O2 sensor so it never gets into bad territory. When the tuner does this, they "write" a new tune to the ECU. A "tune" in this case could generically be called a "remap", but the tuner could do this without even touching how the engine runs. The tuner would read the stock tune directly from the ECU, make the minor changes so the ECU thinks the cat is running within the "good" parameters, then write it back to the ECU. This takes specialized software and usually license tokens (for most aftermarket tuning software ... not sure about the Hondata stuff).

As @EᑎOT stated, it is far easier to just put a dummy O2 sensor in place than it is to put a new tune in the ECU. It's also a lot cheaper ... a LOT cheaper.