Is it correct to abbreviate etc as &c?

I recently asked someone why they had several instances of "&c" in their writing, to which they told me that &c is another abbreviation for et cetera. I have since looked it up and understand now where this comes from, with the "&" being more than just an "and" symbol, that written correctly it is actually based on, and looks like, "et".

Apparently some people have used it this way before, but "some people have done it, especially in the past" makes it neither correct nor accepted in general. One site I came across in my research on this even states that you should not do this.

Finally, ‘etc.’ should not be written with an ampersand instead of the ‘et’ part (&c) except when an older source is being duplicated or transcribed.

That is from a description of when to use etc. from proof-reading-service.com

Is "&c" a correct way to abbreviate "et cetera" in English? Or was it a temporary slang term, similar to "aint", but which never caught on and is used to be quaint?

I wouldn't call "&c." a slang term, given that even the Oxford English Dictionary used it in 1884 (see page 5 of this excerpt from the OED web site):

Hence, while the senses are numbered straight on 1, 2, 3, &c., they are also grouped under branches marked I, II, III, &c., in each of which the historival order begins afresh. Subdivisions of the senses, varieties of construction, &c., are marked a, b, c, &c.; subdivisions of these, which rarely occur, (a.), (b.), (c.), &c.

Typing "&c" as the search term into Google Books produces many more examples of its use in serious text books (Google Books link).

In its day, it was a normal abbreviation of etcetera. Only more recentry has "etc." taken over from it.

Eh.... depends?

An abbreviation is a shortening of something (a(b) - brevere; to make brief). Etc. spelled out is et cetera.

Et cetera is composed of et ("and"; French still retains this form with an unpronounced "t", Spanish drops the final consonant and just has y) and ceter-a, the plural of ceter-us, meaning things (plural) left behind, or remains, perhaps related to the Latin cēdere "to go, move away, withdraw, yield."

So we have etc., where the space is dropped and the letters for cetera are dropped and represented by a dot .

Technically, we have a symbol for "and" and technically et=and. So, with respect to transitive relation, sure: &c., why not? There is precedent the use of numbers or symbols in abbreviations. Take for example "W3", "EC2", and neologisms such as "LGBTQ+" .

I would suggest though you leave it as a plain "etc." (dot included). "&c." seems like an affectation.