Need some explanation

you need to screenshot because we can't actually see your account page

Wow, you have the same account name as me! Games too!

I never noticed that, but now that I look into it... Well, that's strange. I have this remove option for really free games, but also for some non-free ones. It looks like every removable non-free items have been gifted or traded, not bought in Steam store or activated from a CD-key. No idea why there is such a removal link.

Hm, you're right, the "Remove" link appears for me on The Ship, Torchlight II and Don't Starve.

All three have been activated from a tradeable gift.

Maybe you could ask Steam support and keep us posted ;)

Edit: Seems only extra copies of 2/3/4 packs are affected.

I can remove Nuclear Throne ... as if :3

It's a thing Valve put behind users suggestion, IIRC. It has to do with some games getting a "remastered" edition and being listed twice in the library. Take Deus Ex: HR for example: there are 2 versions in my library, one is the "old" version (which is not sold anymore), the other is the director's cut. The Metro series is another example. Previously you just created a category called "Obsolete", put those games in and collapsed it to make your library tidy. Now you can just remove them.
Edit: nope, looks like I'm wrong, the games I can remove are only F2P and some other game.

look a bit closer, I see that those "removable games" are activated from a tradeable gift (as TheDopefish said), but those gifts must come from a 2/3/4-pack. My Tom Clancy's Splinter Cell Conviction - Deluxe Edition, Sang-froid, Red Alert 3 and some other games which were activated from a tradeable gift cannot be removed, because they are single copy from the beginning.

old news