"On the one/other hand" vs. "on the one/other side"

Solution 1:

It's not a matter of "legal" or not, but hand is far more common in OP's construction...

I doubt it's meaningful to explain this as anything other than an accident of linguistic history and idiomatic usage - people tend to repeat the form they hear most often.

Solution 2:

"On the one hand" is more commonly used in English than "on the one side," unless you're referring to legal arguments, e.g. when attorneys are representing opposing sides in a legal matter. English is a Germanic language. In German, "on the other hand" is "anderseits" and "auf der anderen Seite." "Seite" is also the German word for "side."