There's room for a wide variety of responses depending on how loosely anyone wants to interpret "brand". Is a region in China that's known to produce a certain kind of tea a brand, since customers can go into a shop and ask for Bohea? Is a silversmith who puts his mark on every piece that comes out of his workshop creating a brand? What's the purpose of a brand - to encourage a customer keep buying the same thing, to emphasize a unified face over an individual producer, to allow an item to be sold on a larger geographic scale? All of these are going to result in different answers.

Some of the first garments/fashion items that were made with labels were shoes. It's not clear to me when exactly the first shoemaker anywhere may have had paper labels printed with their name and address, but American shoemakers began to do so in the late 1760s (and British shoemakers had been doing so for some time before that). The Townsend Acts were passed at that time, placing taxes on goods the colonists had to import from Britain, and in response there were widespread boycotts and protests; while women had limited abilities to participate in the organized and sometimes violent unrest, they were quite active when it came to boycotts and grandstanding about domestic production. They famously held spinning parties and made skeins and skeins of linen and wool yarn/thread (we don't know very much about the quality and/or whether it ended up becoming usable fabric), and they made a point of buying domestic finished goods when possible. By adopting their own labels, American cordwainers were able to set their products apart from the despised imports and advertise them more effectively to local women. But is this branding? Is just letting you know who made the shoe you're buying and where their workshop is enough to count as a brand?

I don't think we can pin down the first time someone named a product something non-generic and kept it for their exclusive use, but Ivory Soap is widely considered to be among the first branded products and one of the first extremely well-known brands. Devised in the 1870s, it hit as advertising was beginning to allow the use of illustrations and as publications were starting to not allow ads for things like patent medicines, which gave the ones that remained more legitimacy and therefore more power. In the early years of the twentieth century, beautiful full-color advertisements were being printed in magazines like Ladies Home Journal and McCalls, and companies were able to trademark their logos, names, and designs, setting up the situation we're familiar with today.

Before this, how did people differentiate products/producers? Largely by buying from the source, or from a trusted retailer. You would make a decision about which butcher to go to or which dressmaker to patronize based on what you know of their products, your experiences with them, what other people have told you, etc. Does this grocer sell flour with bugs in it because he doesn't take care of his stock? Does that dry-goods merchant mark up his wallpaper to a huge degree?

Potters' marks come to mind as an ancient example of something we may think of in terms of "branding." However, the actual function of these marks remains debated, as many consider their purpose related to the manufacture of these pots, their destination, rather than communicating the origin or maker of the potters to consumers.

Good question. The branding remarks below ring true, and I'd add that painters tended to have studios or workshops which would churn out their pieces or works carrying their particularly style. This at least aligns to the ideas of consistency and projecting a certain image that we associate with effective branding.