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

I always thought that 'on the one side / on the other side' came into English fairly recently because of the influence of Indian languages. Or perhaps it has merely occurred more frequently for that reason.

Tamil: irendu pakam = (on the) other side

"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."

In my own opinion, two pairs are different to use. Your hand only has 2 sides, right? So 'on the one hand' and 'on the other hand' should be used to mention about 2 opposite ideas in terms of an issue.But if you use 'on the one side' and 'on another side' to mention about something, it means that the problem maybe has more than 2 opinions; and it doesn't need to be contrasting. A good example for this is a cube with 6 sides. And that is also the reason why you should use 'on the other hand' but 'on another side'. But I indicate that 'on the other side' and 'on the other hand' are just the same due to the use of 'other' here.

I would say that "on the one hand..." is a natural expression, used normally to indicate contrast: e.g. On the one hand that city is expensive. On the other hand, it's beautiful.

"On the one side" is not a natural expression to indicate contrast, in my opinion. It is a direct translation form Spanish, for example, which is where some doubts may arrive. It can be used to indicate position though: e.g. He left his bike on the one side of the road which was free...".