grammaticality - Why can't we use the plural in "Mt. Fuji is higher than any other mountains in Japan"?

The issue is what are you comparing.

If you are comparing the mountain to each of the single mountains, it is

Mt. Fuji is higher than any other mountain in Japan.

If you are comparing the mountain to all the other mountains in Japan, you need to add an of the and then you get plural

Mt. Fuji is higher than any of the other mountains in Japan.

About your shoes:

(A pair of) Shoes are plural in any case unless he lost one.

He lost a shoe and does not have any other to put on his left foot.