How good is gog galaxy in terms of achievements? Are achievements for the games usually implemented? If you were to compare with the other , the most popular PC platform, is it on the same level or at least coming close?
I am considering moving to use galaxy and this is one of the important factors to me.

You should see on the store page of each game if a game has Galaxy achievements.

If they are there, they usually work.
However, I have two games where for some reasons some early achievements didn't register (so I have: "Find all 10 MacGuffins", but I don't have "Find your first MacGuffin".).

It depends a bit how the game is implemented and when the achievement is registered to Galaxy.

Look at the Charts section of the broken and missing achievements spreadsheet.

Last time I looked into it, about a third of new releases had launched with no achievements. Sometimes on release, "Achievements" is missing from the game's list of features, or achievements are implemented but invisible, and these get fixed after a while. Other times, achievements get added later on.

"How good is gog galaxy in terms of achievements?" --- Absolutely terrible!

Many/most GOG games that have Achievements on Steam, either they are missing Galaxy Achievements entirely solely because the devs love to treat GOG customers like second-class and can't be bothered to do an hour or so of work to give GOG customers feature parity...

...or they technically have Galaxy Achievements, but the Galaxy Achievements they have are broken and buggy and don't actually work, so they don't really have them. And when this happens, neither GOG nor the devs ever actually fix the broken, buggy, non-working Achievements (with maybe a few exceptions of good devs who do once in a blue moon, but that almost never happens).

...and often times, GOG games have some of the Achievements that the same identical games on Steam have, but not all of them.

And these problems are compounded by the fact that probably about 60% or more of the GOG customer base are vehement Achievement-haters whom proactively cheerlead for devs to treat GOG customers like second-class citizens by ripping off GOG customers with feature-removed, Achievement-less games on GOG, even when those same identical games do have Achievements on Steam.

This problem is also further compounded by the fact that many/most GOG customers continue to buy the rip-off, inferior, feature-removed GOG versions of those games, which just further emboldens & encourages publishers and devs forever to continue ripping off GOG customers with second-class citizens, feature-removed, Achivement-less GOG versions of their games.

TL;DR: In summary: the GOG platform is absolutely horrendous when it comes to Achievements.

Speaking unbidden for the haters, I don't "cheerlead" anything, I just don't give a rat's ass either way about the moronic achievements nonsense that would only work with a client I detest and refuse to use anyway. I really could not care less if the devs give you that infintile pat on the back for every five steps made in the game or not. Just don't try to rob me of DRM free games just because they don't have a completely superfluous feature that you consider all important.

Not exactly a hater, but I'll never let achievements determine if I buy a game. But I think I speak for a sizable contingent when I say this: If achievements are really SO important to you, then go buy on Steam.

Power to you, but I consider that "incentive" to be no incentive whatsover. Either the activity itself is fun, so I don't need a gold star and a pat on the back for it, or it's not fun, and then no virtual trophy will make it worthwile. It's all patronising, empty ego-stroking.

I have achievements when I play on my PS4, and I have never felt compelled to go out of my way to get any, or gratified by any of the ones I got. It's all an antithesis of just having a fun time playing, immersing myself in the game. And at least half of achievements for every game seem worthless. I think on average I get about 50% of achievements on a finished game, just by playing it and never trying to get one, and trust me - I'm not a very good gamer. If I can get it, then it's no achievement :D

Sometimes you don't even know that you can do something without the achievements, mostly if there are several ways to solve a problem. Last one doing that for me was the "pacifist" achievement in Born Punk.
You can just shoot your way through or you can find more creative solutions. If you don't fire a single shot until the end of the game (it's just three choices really), you get the pacifist achievement.

However, I would be perfectly happy if achievements would be only ingame. But not many games have these ingame. It became standard that they are logged outside the game.

Of course this can also be exaggerated. If you have five characters to chose from and you get a achievement for every pair of chars you finished the game with, that would be 20 combinations and that's a no go for me.

In Lego games I don't really need achievements, but I like to see the 100% counter :)

I get that... but is that a good thing? People used to try stuff out because they didn't know. Because they were invested in a game, because they wanted to see what will work and how. And the results felt so much better for it. For example: when I managed to get out of that ambush in the first Deus Ex, the one you're not supposed to beat and the game will only actually continue when you get captured, that was all me. No one told me to do it, no one told me if it even can be done. It felt like a triumph to pull it off in a way it wouldn't if it was something I took from an achievement list. Sure, there's probably plenty of stuff I missed, but I think it's a fair trade off for how good all the stuff I did pull off felt. That's the irony of it - because of "achievements" things like that feel much less like achievements. They are just things on a checklist.

I'd say yes. There's still plenty to sniff out without getting an achievement.
And there's usually lots of hidden achievements too, so you don't even get a spoiler when checking out the list.
Back in teh days thre were not so many games around, I must have played Indy 4 hundreds of times (literally). But these days? Games have become bigger and I have many many many more of them. I hardly play anything twice now, I am glad if I finish some of them. So yes, seeing that something is possible might be a good motivation to play it again.

No one is forced to go for achievements, I play most games without, but in some cases achievement hunting it's actially fun.

Problem with ingame achievements (you can find a list here in case you did not know) however is that you can't turn off the achievements and in case that an achievement also shows a popup during gameplay it might become distracting or immersion breaking for some. I never saw any game with ingame achievements that also offered the option to turn them off for those who disliked the popups ... so imo they might have a downside as well.

Monkey Island 6 does not have popups, but it has ingame achievements and I have seen at least two more games that behave the same (probably more, but I usually don't pay attention if there are achievements in the game or not).
Popups in offline games are quite rare in my experience. Right now I can't think of a single one. I am sure that I have seen one, but I really can't remember which one it was.

You are correct. Popups in offline games are extremely rare but they do exist. "dv Rings of Saturn" i.e. and iirc "Ghost 1.0" and "UnMetal" have them but yes, since they are so rare we can agree that the downside of ingame achievements therefore is also quite minimal.

There are so many games to play and there is always a question: when shall I move on to the next one? Achievements can be a good guideposts (or a checklist) giving a hint if I have experienced the most relevant content and faced the most interesting challenges.
This can be especially significant for games of non-linear nature, games with multiple endings or with some 'consequences', etc.

Thank you for the answers to the question. I can see that there are many achievements missing in galaxy, also for games which are of interest to me. Let's hope that the situation will improve! Regardless if you play with achievements, or you create your own in your mind, or you don't care at all, I think that we should always ask for the best all-around product!