I bought JJ2 Secred Files and Christmas Chronicles. In both games after intro videos, main menu shows up but game reads some key pressed (like up or down arrow) so there is a "selected option" loop in main menu which makes it unusable and i cant even start new game.

It only happens in those 2 games, havent notice any issues in any other games i have/had
I tried to disconnect all my controllers - X360, Xone, Saitek pz31a, and turn off Corsair keyboard. Nothing helped

Are you using the original, Jazz2+, or J2R?

Im experiencing this as well. Using the version provided by GOG. It seems to spam left input. I can get into the game by "playing roulette" as the input makes the menu unusable. When I did get into game, the character would be moving left constantly. When I unplug my keyboard it stops.

*** EDIT ***

I resolved this issue by selecting "JJ2+ Mod" in the installation manager on GOG

Pretty new to GOG, but I can't seem to find the JJ2+Mod option anywhere.

I purchased the Jazz Jackrabbit 2 Collection, and I'm having the same issue with being unable to make a menu selection.

I have been looking in my library>JJ2 Christmas Chronicles>Settings button>Manage installation, but I have been through all the options, and don't see "JJ2+ Mod"

Am I looking in the wrong location, or is this for a different version of the game?

Right location, wrong game: you must look under Jazz Jackrabbit 2: The Secret Files. Read here.

The mod doesn't apply to Christmas Chronicles, however, according to this post, you can play its levels in the modded version of the game.