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?

Most Public Radio stations are at or near the bottom of the FM band.
Although this could be some freaky overload problem, you're most likely
hearing (say) one of the news programs on two different stations. The way to
find out is to listen for their call signs.