Yes, but you can also "climb down" so without context, the preposition is important
I see your point, but can you make that argument in other than English? Climbing implies upward movement, for example, climbing the ladder of success. For most, the expression does not imply "failure." I feel that "climb up" is literal, while "climb down" is an oxymoron. The opposite of "climb up" is "come down."
Prepositions are arguably the most difficult part of learning a new language. It doesn't matter if climb and climb up convey the same meaning in English. What does matter is learning how to say up, down, inside, outside, and so on in the object language.
Climb on and climb on top are the same. Can you climb on something without being on top? Or is it "climb on the hill, but never reach the top?"