Much like today's Chanel or Gucci, did ancient people have any products made by a certain person or group that were branded? I know the rich used gold and jewelry to show off, but did anyone say "Yeah, I like all of these things, but I think this person/group does it better. It's cool to have their logo on my stuff."?
This is maybe a little past "ancient times", but you may be interested to hear about a certain Frankish sword maker in the 9th century!
Throughout much of the early Middle Ages, Western Europe was dominated by the Carolingians; that is to say, the legendary Frankish king Charlemagne and his descendants. Their territories, though not always united (Charlemagne's empire broke up after the death of his son in 840,) included at their greatest extent much of modern day France, Germany, and Italy. These large realms, containing so much old Roman territory, had relatively advanced technology and a strong trading presence throughout Europe. It is no surprise, then, that Frankish goods are found at archaeological sites throughout much of Europe.
In particular, Frankish swords, which resembled a midway point between the high medieval arming sword and the shorter Roman style of weapon, were seemingly ought after throughout much of Europe; Arab writers noted that even Vikings and Russians wielded swords of Frankish make. Relevant to your question, however, is a sword-maker by the name Ulfberht, a name generally traced to lower Francia, though historians have never been able to pinpoint where exactly Ulfberht lived or sold his weapons from.
We know Ulfberht's name because, as an apparent seal of quality, a number of Frankish swords are inlaid with the inscription "+VLFBERHT+" (the letter "U" being written with a V at this time.) Ulfberht swords have been found from, as Simon Coupland puts it, "Iceland to Russia", and there is a "paucity of finds from within Francia itself", indicating the extent to which these swords were sought after as a trade good; in particular, they have been extensively found as Viking grave goods.
Ulfberht swords were seemingly so well known that some weapon-smiths supposedly even pirated them, putting the Ulfberht inscription on their own inferior-quality blades in an apparent attempt to cash in on the brand. Though the Carolingian rulers extensively legislated the arms trade, copyright law was, obviously, an unknown concept to them.
Coupland, Simon. "Carolingian Arms and Armor in the Ninth Century." Viator 21 (January 1, 1990.) http://search.proquest.com/docview/1297909336/.
Winroth, Anders. The Conversion of Scandinavia: Vikings, Merchants, and Missionaries in the Remaking of Northern Europe. New Haven: Yale University Press, 2012.
Books created with paper, whether manuscripts or printed, have had identifying watermarks from at least the 13th century (1), which help to establish when and where the paper of a particular book was created, which could be likened to the concept of a brand. Ground woodpulp would have been soaked in water, then raised upwards with a wire frame and then pressed with wooden boards to flatten and dry, what would become paper. An effect of this is that on the paper, the imprint of the wire frame as the moist paper was lifted upwards can be seen in the final product; papermakers went to the effort to create elaborate designs (fruits, animals) in their wire frames as a way of identifying the papers; hence the term, watermark. Sometimes the watermarks are incredibly subtle: the Royal Library of the Netherlands has a nice article on the application of Electron Radiography to reveal them (2). The National Diet Library of Japan has an accessible introduction to early modern printed books (anything printed before 1500 is termed an incunable (incunabula for plural), which also discusses the history of watermarks in such books.
Notes:
(1) The National Diet Library of Japan, 2004. "The World of Watermarks" in Incunabula. Available at: https://www.ndl.go.jp/incunabula/e/chapter3/chapter3_02.html
(2) The Royal Library of the Netherlands, 2012. Watermarks in Incunabula printed in the Low Countries (WILC). Available at: http://watermark.kb.nl/page