When referring to a restaurant specializing in fish and chips would you call it a fish and chip shop or a fish and chips shop?
"Attributive nouns" or the first elements of compound nouns tend to be singular in form
There is no absolute rule forbidding the use of a plural noun in the first part of a compound, but it is more usual in general to use the singular form.* "Fish and chip(s) shop" doesn't seem to be an exception to this tendency. The Google Ngram Viewer suggests that both forms exist, but that "fish and chip shop" is more common than "fish and chips shop":
I don't think phonetics is an important factor
Some comments have brought up the supposed phonetic indistiguishability of "fish and chip shop" and "fish and chips shop", but I can't see how that could determine the spelling one way or another: even if it is true that nobody ever pronounces these any differently in practice (which I rather doubt), that wouldn't prevent people from using the spelling "fish and chips shop" for the pronunciation [fɪʃn̩t͡ʃɪpʃɒp]. I think the use of the spelling "fish and chip shop" is based mainly on grammar, not on phonetics.
*Some exceptions to this tendency are mentioned in the answer here: Singular/plural Nouns as Adjectives
When a noun is used as an adjective, it is almost always the singular form, even when the noun is not normally used as a singular. E.g:
- car - car alarm
- house - house key
- trousers - trouser press
- spectacles - spectacle maker
- clothes - clothes line
So, "fish and chips" is the noun phrase, and becomes singular:
- fish and chips - fish and chip shop