It is not everything you want, but from SEDE you can get:
- Score of the synonym. (You still do not know how many people voted - but at least you know the difference between the number of upvotes and the number of downvotes when the synonym was approved.)
- The user who suggested the synonym.
- The user who created the synonym. (Which will be the Community user if the synonym was created by voting of regular users. If it was approved by a mod, you'll see the name of the moderator.)
Here is a query listing approved synonyms - ordered from the most recent one. Here is a query showing data about synonyms for the given synonym - as examples, it is applied to code, https and rss.
If you're looking into this because some synonym seems unsuitable (and possibly needs to be removed), it is useful to know that from SEDE you can get separate lists even for synonymized tags. Example: code and code-formatting. (So you find the questions which had the "old" tag when the synonym was created and the tag hasn't been changed since then. However, the synonymized tag is changed to the master tag if a question is edited after the creation of synonym. And all occurrences are changed if the two tags are merged by a moderator.)
You can check out the entries available in the TagSynonyms table here: Database schema documentation for the public data dump and SEDE.
Since we're looking at the tag synonyms, here is a query listing the pending tag synonyms. IIRC the list on the site will only show those pending synonyms where you can actually vote.
Just as a curiosity, I will mention that the moderators have the possibility to suggest a synonym in such a way, that the synonym is still left for other users (regular users or moderators) to be approved: Synonym by a moderator which needs approval?