It gives no result, even by clicking on a tag at a shoot-em-up game.

Cannot really search anything is my sad experience though finding games by their titles seems to get better.

Looks like GOG has only tagged a single game as Shoot 'em Up (Blazing Star), even though there should be several others. GOG's genre tags leave much to be desired ...

Gog tagged likely all shoot 'em up as shoot 'erm up, like AngerForce:Reloaded, and whatever. But while clicking on other tags (action, Point-and-click, etc.) works, shoot 'em up does not give a single result.

Hm, you're right. I thought clicking on it didn't work because there were no other games with that tag, but it seems Blazing Star and AngerForce: Reloaded share the same Shoot 'em Up tag. There are also other shoot' em ups that are just tagged as Shooter though, but that doesn't explain why this particular tag doesn't work. Weird. You might want to contact Support about it.

I can't speak for GOG, but it appears that the presence of either the space or apostrophes in the tag is causing it to be entirely excluded.

Searches for shoot return entries with shooter, but I could not recognise games tagged as 'shoot 'em up' in the list. (I am happy to be proved wrong though - I didn't check every game)

I believe that the shoot 'em up tag is unique in having either a space or a single quote character in a tag name.

However, the search function is capable of of dealing with both types of punctuation (Anna's Quest [1])

The games in question are part of the index, and can be searched for either by name (eg Monolith) or by one of the other tags (eg ComBat). What I have noticed is that you can have either a title search or a tag search, but not both at the same time. ("Sci-fi" and "X-Wing" will return Star Wars games. "Sci-fi X-Wing" will not). You can't match multiple tags either "shooter sci-fi" will not return any results, despite at least one X-Wing game having both tags.

For this to make sense from a database perspective, the fields being searched (title and tags) must be handled separately. A search will query all titles that match, then all tag1s that match, then tag2s, then tag3s and then merge the results.

I would suspect that whatever intelligence was added to handle the titles with apostrophes and spaces has been omitted from the tag query. (For SQL databases, special handling is always needed for apostrophes as a protection against hacking - see SQL injection attack. Perhaps the tag indexer performs this protection by dropping the tag entirely - the bit before the objectionable character doesn't get indexed).

Despite all that, the best workaround I can provide for you is the following google search
site:gog.com "shoot 'em up" inurl:game

Note it will also pick up games with the term "shoot 'em up" in the description or reviews, not just the tags.
inurl:game is present to exclude announcement and forum posts.

[1] https://www.gog.com/games?search=anna%27s%20quest&page=1&sort=popularity