Dunno, Merriam-Webster list it as both,[1] though from your comment I get that it is most commonly understood as the store. Thanks for the tip though, I was only aware of the food meaning.

[1] https://www.merriam-webster.com/dictionary/delicatessen

Perhaps it depends where you are, Oxford English Dictionary only mentions the shop meaning.

Weird! :-O

This is what I get:

1: ready-to-eat food products (as cooked meats and prepared salads)

2: sing, pl delicatessens [delicatessen (store)] : a store where delicatessen are sold

It's fine as a modifier ("it's a delicatessen salad") but not on its own ("it's a delicatessen")

Delicatessen is also plural in German if you use it to mean "delicacy", instead of an adjective ("chai tea"-style).

False. It's fine as a noun if you leave off the article, which OP did.

The "ready-to-eat food products" meaning surely comes as an abbreviation of "delicatessen-style".

It's a wonky marketing term like "lite".

The store is usually shortened to deli where I live.