Product catalog database

Also, this doesn’t need to a be specifically a product catalog. This could be a photo gallery that is used on the “products” page, if I could define tags for each photos and narrow down all the photos to specific ones based on which tags were clicked.

You will probably have a hard time finding a plugin that will accomplish what you want. I am trying to think of ways to do the same thing and the only options I can see, are creating individual pages for each product which is fine as long as the product list is not too large. The upside is you can format each page however you would like and display as many images as you please.

You can also use an ecommerce package and disable the pricing information. You may or may not get all the functionalities you want but it will likely be more simple than lots of individual pages.

Or use posts, with tags or possibly custom fields (use a plugin such as https://www.ads-software.com/extend/plugins/custom-field-taxonomies/ if using custom fields).

I haven’t tried these but these may help you with what you want:
https://www.ads-software.com/extend/plugins/tags/product-catalog
https://www.ads-software.com/extend/plugins/eshop/