At least in Classical Latin, as used in late Republican / early Imperial times, it is a consonant. And quite distinct from the vowel. In fact, that "Ⅎ" I mentioned was an attempt of some emperor to spell /w/ [w~β] apart from /u u:/ [ʊ u:]; that wouldn't happen if at least some speakers didn't catch "hey, /w/ doesn't look like a vowel".

But before that, it's way more complicated. The Etruscan alphabet had a letter for /w/, it's ⟨F⟩; but when the Latin alphabet popped up, instead of using it for /w/ they repurposed the letter to /f/ [ɸ~f], and decided to spell /w/ with the same ⟨V⟩ as /u u:/.

Going further back in time the things get even messier, as PIE *u is clearly a syllabic allophone of *w, not its own phoneme. So... "is /w/ a consonant or a vowel?" "yes".

After that (the Romance languages) it depends a bit on the language, but I feel like the mess is back on the menu. That Latin /w/→/v/ is a consonant, sure, but other instances of [w] popped up; either as part of a vowel or as /u/ forced into a syllabic position. For example, in Spanish I'm pretty sure it's being handled as a vowel, because:

• it pops up in vowel alternations, like /o/ vs. /u̯e/; see ⟨dormir⟩ "to sleep" vs. ⟨duermo⟩ "I sleep". And typically when you can sub one thing for another, in language, those things are being handled by the language as part of the same category.
• if you analyse Spanish as having /w j/, you'll find the only occurrences of onset CCC demand either; like ⟨prueba⟩ /pɾweba/ [ˈpɾwe.β̞a] "proof". It's simpler to analyse them as /u̯ i̯/ instead, then claim Spanish allows up to two consonants in onset.

It's less ambiguous than your English example with "raw", where the absence of linking-R you mentioned hints it's already a consonant.

Honesty, one motivation for writing the comments is that I’m glad there is activity in this community and I want to push it. I could use more linguistics in my life so thanks for engaging with me!

You're welcome!