Synaesthetia is a well documented phenomenon, it's inducement from chemical agents is less well understood, but there is a large body of evidence that supports it's existence. See this metastudy for some examples:
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3797969/
Out of interest, how many senses do you think you have?
Depends on what you classify as a "sense". If you want to count hunger as a "sense" then I'm not really sure, not something I've contemplated. Again though the descriptors I talk about when people speak about their experiences aren't in concordance with synaesthesia, because their experiences are never similar regardless of dosage we partook. I would be greeted with mostly paranormal descriptors like "I can feel the heat from the boiling water in the kitchen" sort of stuff. And things like "I am a race beyond this star system that uses smell that hops from one atom to another to communicate"
You know, that sort of stuff.
Associations with color aren't of interest to me for example. In Asia you have folks being terrified of the color white (somehow linked with death experienced in hospitals and whatnot) while black would be the usual color for death here in the West. The folks I've been with never have reproducible experiences. It's always something new, and always a loss of any sort of cognition according to them during and soon after the come-down.
When I hear treble notes for example, it can sometimes be akin to "sweet" in a slight sense. But I don't think I'm experiencing something sweet with my taste buds. Mostly an association of sorts with old ads of sprinkled powered sugar over foods with high pitched sparkle in the background.
But seeing unicorns and communicating with them even though I was right there looking at the person, while they're telling me the unicorn is in the wall, that can move, and can't move within it.. That's just incoherent ramble.