I don't understand this sentence. Does it mean that Elvis had poor taste? Is "brash" a negative word? "Elvis's tastes were brash, rather like those of many other ordinary unsophisticated folk."

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Here is the definition from Longman Dictionary:  "a brash building, place, or object attracts attention by being very colourful, large, exciting, etc.  
Example: The painting was bold, brash, and modern."
(https://www.ldoceonline.com/dictionary/brash)

It seems that the word is used in a disapproving way in your sentence above. For example, if you look up photos of Elvis Prestley from performances in the 1970's you'll see that the clothes he wore for his performances were really "flashy". If you say his outfits were brash, you disapprove of his style.

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Brash can be used for something "tasteless appearance"
And in this sentence brash is a negative adjective.

Hope that helps.