I have an clock radio about 30 years old that, when tuning for 88.5
FM, for example, by turning the dial, will find that station, and then
a little higher will be another station, and a little higher still
will be 88.5 again, often a stronger signal than the first one.

I'm sure it's not a second station playing the same stuff.

I know about harmonics on AM MW and Short Wave, but here the tuned
frequencies are so close, and also I didn't think there was stuff like
this involving FM. What is going on?