Hi Kim and welcome to the forum,
You can sell as many different products from as many different categories as you want to, there is no limitation. The up side to doing that is that you are covering a wider market and increasing your chances of sales. The down side to it is the cost as the more items you list, the m ore listing fees you have to cover, sale or no sale, so you always have to be very mindful of that.
To sell under a single brand or break the brands up, that can be six of one and half a dozen of the other, everyone has their own preference. I've never seen any documented evidence that has been based on "what I would consider reliable data" showing that one is better than the other. I think it comes down to a smart layout design and good cross marketing.
For example, if you are going to sell everything under the one brand, then you have to be careful not to let it look cluttered. Having dog shampoo for sale with jewllery, along with paint supplies all for sale together can make the store look like a dollar store and can devalue your items. You need to clearly define categories for customers. Think of a variety bricks and mortar store, Walmart for example. Lots of variety, but all well defined in their own areas. It's no different to selling online, the same basic layout principles are in play.
If you break your brand up and sell individually, then it gives you a much cleaner layout and opportunities to really define branding that relates to each genre, however it's important to link your marketing. For example, one of the BIG benefits of selling off the one account is that you are drawing a cross section of buyers into the one store. Someone may come in looking at the dog shampoo, and end up buying some paint supplies as well while they are there, despite the fact they didn't even know you had those products when they came in.
Now you lose that marketing power to a large degree when breaking your brands up, so cross marketing becomes really important across all of your platforms. For example, let's say that you choose a lead brand name like "Kim's Variety' Boring I know, but let's run with that as the example.
Now for your paint supplies store, you would then use your lead brand "Kim's Variety" along with a sub-brand, so perhaps "Kim's Paint Supplies" What you are doing is reinforcing your lead brand all of the time, but still defining your individual stores. The lead brand is the one you want to burn into customer's minds, that's the one you want them to remember.
That's part of cross marketing, but you also have to ensure you complete that by having a little spiel about your other specialty stores and have links to each of those across all your stores. That then makes it easy to draw customers into those other stores, no matter which store they come into.