A tag shop=dairy is used for shops selling dairy products. Not rare such shops are not big stores, which are designated for inhabitants of some urban quarter, which get new fresh production every day.
A production facility or factory producing dairy products should instead be tagged using man_made=works with dairy specified as the product=*.
In New Zealand, "dairy" usually refers to a small convenience store selling several kinds of basic food items. These should be tagged using shop=convenience, not this tag.
How to map
Set a node at the center of the shop or draw an area along its outline.
Tag it with shop=dairy.
Tags used in combination