Firefox tried to use it and was sued for trademark infringement. wxPython also has a Phoenix project. It's definitely a catchy, but overused, name.

I don’t think the problem was with “phoenix”.

It was Project Phoenix (resurrection of the Netscape browser). This resulted in the Firebird browser (Firebird and Thunderbird). But Firebird was an existing database that objected to the name. So, we got Firefox instead.

At least that is how I remember it.