How to change the default browser when using the "look up" feature?

No, that's not it. The "lookup" feature defaults to the macOS Dictionary, but if it comes up empty its next choice is to search Wikipedia. If that fails I don't know what happens next.

In any event it's not a question of browser choice; it's just the way the lookup feature happens to be implemented in macOS.

In case you're interested:

That screenshot was from this Discussion using Firefox. Its default Search Engine is also DDG. But yes, the "See More ..." link opens Safari.

And in fact it's app-specific. In the course of researching your question I managed to make the Notes app to become unresponsive upon looking up that word. I tried another word, same thing. Now I'll have to file a bug report.

In the course of researching your question I managed to make the Notes app to become unresponsive upon looking up that word.

It only occurs with one particular Note. It also happens to be a very long one. I surmise Apple won't care.

Do I need to change the default browser for just the "look up" feature somewhere else, or what?

I may have misunderstood your question. If you change your default web browser to Firefox, clicking the "See More..." link in the above screenshot will open Firefox.

System Preferences > General > Default web browser: choose.

Cevan77 wrote:

Yes, but if it doesn't find a definition or anything and you click the button to look for it in your search engine, it opens it in Safari as opposed to Firefox.

Not if you change your default web browser to Firefox. If it's still using Safari after doing that I have no explanation.