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"Mix a few tablespoons of baking soda with a cup or two of hot water"

Yeah! That creates an alkaline solution, and if you pour that down your drain, it can cause some fats trapped in there to turn into a soap, which is water soluble, helping you remove them.

"(you can also add white vinegar for a bit more punch)"

... aaaand you failed high school chemistry. Vinegar is sour and neutralizes the alkaline solution. Worst case, the reaction causes heat and bubbling that may damage your pipes.
baking soda is a base, it's alkaline. vinegar is an acid, it's acidic. adding an acid to a base makes it less alkaline.