Can I use different logos on different products?

Short answer: Yes. You can upload as many logos as you like and freely assign a different one to each product.

Common use cases

a section, a team or a department of your organisation has its own logo;
women's or junior products carry a different variation;
a sponsor logo on a few products only;
one logo for light products and another for deep products (see Light background or deep background — which should I choose).

Upload each logo separately from the “My logos” tab — see How do I upload my logo.
Give them short, distinct names (“Red Logo”, “Youth Section Logo”…): that is what will let you find them quickly.
On a draft of the shop, click the product in its collection to open its configuration, then, next to the placement you want, click the thumbnail of the logo to apply — see How do I put my logo on a product. All your logos are offered on every product: nothing ties them to one another.

Many organisations do not realize this is possible and limit themselves to a single logo for the whole shop.

You'll know it worked when: Each product carries the logo you assigned to it, independently of the others.