That is at least somewhat logical, if not a bit overly paranoid. A microwave can cause damage if the shielding is damaged, wifi cant ever cause damage.
It could….
If you stuck your head next to a router 24/7 for years with it blasting at full power.
Most consumer grade routers (and enterprise grade access points for that matter) are unable to produce more than 1W. Even with a higher gain factory antenna you might be radiating maybe 30W. There is no conceivable length of time that you could be exposed to that radiation and suffer ill effects from it. I’d be surprised if it was even enough heat the air around it.
No, radio waves are not ionizing. (Unlike, for example, ultraviolet or X-ray.) Ionizing radiation can cause cumulative damage, because each photon quanta has enough energy to potentially change organic molecules. But low frequencies such as radio waves, (anything lower than visible light) can’t change your molecules. The most they can do is heat you up, just like visible or infrared light. So unless the radio transmitter is high powered, (such as a microwave) the radio waves won’t do any more than the lightbulb in your room. I’m assuming you don’t live in a dark cave.
Most microwaves (especially old ones) are shielded very poorly. However microwave radiation is nonionizing so the only harm it is going to do to you is burns if you get hit by enough of it. Needless to say you aren’t going to get hit by that much no matter how poorly shielded your microwave is. The worst any consumer microwave will do is screw up your wifi reception around it.