Tag synonyms show up in the drop-down list of possible tags, which is confusing my site’s users.
I would like them to:
… but never be presented to users as a suggested valid option.
That would break totally my tagging and after that no one on my language family would use tags anymore. There are languages beyond germanic ones 
So I disagree. Synonyms are there because we have different words meaning same thing, and people have tendency to use and therefore search different words aka. synonyms. If they didn’t find some spesific term they don’t start over and trying some other synonym.
That couldn’t be a sitewide policy because — and now I go to harsh examples — every languages aren’t that poor there is just two words for happiness, one word for snow or mobiles or just one dialect by vocabularity — Anglos are counting english(US) and english(UK) as two separate languages (oh you poor aussies, kiwis, south africans etc…), but those aren’t, they are just using different synonyms.
I don’t understand the last part of your post, but the first makes sense. I’d be happy with a “Hide tag synonyms from dropdown lists” option.
Out of curiosity, can you provide some real-life examples? 
To add a bit on the first post, synonyms show up on the autocompletion as well:
In search:
In post:
Is this really a thesaurus feature built into the search bar, or the results are just showing words based on what letters are being searched for?
I can see how this would be confusing if it’s really an unlabled rogue thesaurus engine running in what looks like just a regular search bar, while that could be useful but probably best to be labled as a thesaurus and be separate from search system depending on the language.
Usually best to consult with the pigeons for synonyms:
Sandwich and submarine. And we finns have totally 100+ words for snow.
You call a sandwich a submarine? There is a sub sandwich, or subway sub sandwich.
That is a lot of different words for snow wow.