To those who finished both games:
I'm still in the first game (20hours in, but only day 9 of the first protagonist) and while I love the atmosphere, the town and its inhabitants and story, I have problems with the (combat) mechanics. This makes it really tedious to play for me.

Therefor my question is: Should I try to continue the first game or is it better to abandon it and play the second game (which hopefully has better control and combat mechanic)?

There is no reason for you not to try the new game.

This is not even a sequel. Don't believe the title. That was the dummiest thing in this game, with this "2" in the title.

No, I know perfectly well, that this is a (Second) remake of the original game.

My question still is, if the new game contains all the important parts of the old game and only modernized the engine and gameplay or if it is a different telling of the same story.

The remaster is not the remake. They put "2" in the title only for non-CIS dwellers and only because of that old American rule about the same titles of different products.

It's a different telling of the same story; the overall story hasn't changed and all the important parts are still there, but there's differences in the way it's told, new content and different perspectives on old ideas. It's a good mixture of old and new.
Currently only the Haruspex story is out and the other two are still being finished, so if you finish the Bachelor you're perfectly situated to jump into Pathologic 2.

Thanks. Then I will soldier on and at least complete the Bachelor first

Even the old game has its charm despite its many flaws, or perhaps even because of them. By playing completing the storyline for at least one character, you'll gain a better appreciation for the new game, even become more adept at surviving.