There are many mountains or an unknown number, and you can climb anyone of them. You could say that you can climb every one of them, but that is limited to the mountains that are known.
Any (one) mountain is a broader range, if you were to discover a new mountain, you could also climb it.
I would not say the reason is that you can only climb one mountain at a time, if we use the example of fighting we have to say: I can beat anyone who challenges me even though, conceivably, we could fight more than one person at a time.
The idea is: there is no one person (singular) that I could not beat, I can beat anyone. There is not a single mountain that I could not climb, I can climb any mountain.
You can say any mountains, but you must limit your range (any one of them) e.g.: "I can climb any (of the) moutnains in this region."
To be clear: You can say "any mountains" when it is followed by reference to a specific group of mountains.
What seems to be causing confusion for you is the "imaginary" reference to "potential mountains". It seems that you are running into this difficulty in situations that are very open and refer to general possiblility.
So let's try this:
You can climb any mountain (whatsoever, that you know of, or don't know of) steb by step.
This use should become more clear as you come across words like:
anywhere, anyone, any time and so forth. These words refer to "imaginary" or potential situations, the thing referred to is not specific but an "open set" and so the singular is used.
You can climb any mountains you come across.
You can climb any mountains in the country, but not outside of it.
This use means "any ONE of them", the "one" has been removed out of habit (unfortunately). This set is referring to a specific number or list of itemS so the plural is used.
But honestly this is a very difficult problem because it depends on how you look at the items, as real or imaginary. For example:
"Any car parked in the lot overnight will be towed."(potential)
"Any cars parked in the lot overnight will be towed."(real)