Today I know people who hold that fish skin is considered waste (whether in their own view or even according to everyone), and nevertheless they remove the skin from the fish flesh.
The question still is: what could be the reasons to permit it in the case of fish skin? Is it because in the case of the fish and the skin, we are dealing with the same object according to their view (the skin is part of the fish)?

In principle, you need to ask them. There is a line of reasoning that peeling a fruit or a fish is necessary in order to eat it, and therefore the peeling is not selecting but part of the eating process. See Biur Halakha, siman 321, s.v. “to peel.”

Regarding fruits, I read on Google that peeling is not considered selecting part of the fruit, but rather removing its outer part. It could be that this is relevant to fish as well.

Let me ask it this way — do you peel fruits and vegetables on the Sabbath, and do you remove the (outer) skin of a fish?

I don’t eat fish, but I do peel fruits.

And if in the future you do eat fish, or someone in your household eats fish, would you tell them (based on the comparison to removing the peel of fruits) that it is okay to remove the skin of the fish?

And if you would rule that it is forbidden to remove the skin of the fish, why is it permitted to remove the peel of the fruit?

You’re hard work. Yes, I’d tell them it’s permitted.

So why are there people who forbid removing the skin from fish if, according to everyone’s reasoning, it is permitted to peel a fruit (or even to peel the peel from a piece of fruit)?

Sorry for being a pest. I want to understand the reasoning in Jewish law.