While this is certainly outside my wheelhouse, what I see in various locations is that (at least for English)

- A multi-word phrase is a phrase, not a word

- A lexeme is a basic unit of meaning in a language, like a word (and it's forms [1]) or phrase.

- Every place I was able to find described a lexeme as a "word _OR_ phrase", making it clear those two are different things.

- Dictionaries, in general, focus on words. Many do include phrases also. This point is less definitive; and just my understanding from looking at dictionaries and how they describe themselves. That being said, every source I can find that discussed something close to the topic seems to support this

[1] A word with all it's forms, in that "walk", "walked", and "walks" are all a single lexeme (with each form being a distinct word) OR a phrase

Side note: I'm not looking to "correct" anyone; just pointing out what information I'm able to find on the topic. I'm open to being corrected, but that correction would need to include reasonable sources.

While not all phrases are words, the specific phrases we are talking about are a type of word known as an open compound word.