Earlier quoted context omitted.
So, 2+2=5 is true in the ring Z/Z (arithmetic modulo 1) which is just the trivial ring consisting of a single element. It's not true in any of the other modular arithmetic rings. So, it's not particularly interesting. If you move away from modular arithmetic and the numbers 2 and 5 being in any way related to their standard meanings, you can of course redefine them to mean whatever you want (e.g. rotations of a cube…
IIRC the moral of the original 2+2=5 guy's comment was "if someone reaches a different conclusion than you, ask them under which axioms they reached that conclusion." I think that's valuable advice. Note that he didn't say you have to accept their axioms. Clearly working in arithmetic modulo 1 is meaningless in most circumstances, and you can criticize that. But if you're not reasoning under the same axioms, you cann…
If so, then the argument he makes is totally braindead. He doesn't even consider special algebraic structures or anything. He's just coming up with the most obvious observation ever: if your mathematical model doesn't match the reality or asks the wrong questions, then the results are worthless. This is probably obvious to any high school student but that's certainly no reason to run around and say that 2+2=5. In any case, there is zero mathematical insight there, he doesn't even make a mathematical argument for it, and the "I am a former mathematician" part only serves to distract from that fact.
I could go onto an entire different rant about his infinity argument and how mathematics seemingly haven't figured out how to work with infinity "naturally" (I don't know what he means by naturally, but they certainly have figured out how to work with it precisely), but let's leave it at that.