To be really technical about it: there's a reasonably popular brand of tape called "Duck" (https://www.duckbrand.com/). They refer to their main product, which is a kind of duct tape, as "Duck Tape." To clarify that the term "Duck Tape" refers only to their company's product, they refer to it as "Duck Tape® Brand Duct Tape."
So the phrase "Duck tape" is fine, if you capitalize the first word and use it to refer to tape made by the Duck brand. Likewise, "Duck Tape" is fine, if you capitalize both words, since it is the name of the specific product line made by Duck.
In fact, if you call store-brand adhesive bandages "Band-Aids," and you call store-brand acetaminophen "Tylenol," you could argue that it makes sense to call store-brand duct tape "Duck Tape." And if you don't always capitalize "Band-Aids," you could argue that it makes sense not to capitalize "Duck Tape" either.
So I would argue that using the phrase "duck tape" to refer to duct tape is fine, provided you accept that it is effectively a genericized trademark.
Fascinatingly this folk etymology is incorrect. According to Wikipedia[0], the “duck” refers to cotton duck[1], a strong fabric that can be made waterproof, from which the original duck tape was made.
“Duct tape” (with a T) is in fact the retronym and was coined when duck tape started to be used for ducting.
All of this doesn’t explain how Duck-brand tape managed to get a trademark on an existing generic term.
To clarify: I'm not claiming that the phrase "duck tape" in its original sense was derived from the branded product "Duck Tape"; as you point out, the phrase "duck tape" came first, and was derived from cotton duck.
However, if you use the phrase "duck tape" not in its original sense, but as (effectively) a new word--as a genericization of "Duck Tape"--then you would (IMHO) be quite correct to call duct tape "duck tape."
I guess this depends on whether you think that this use of "duck tape" is actually a new sense of the phrase, or just a continued misusage of the preexisting sense.
I would argue for the former, since IIRC the phrase "duck tape" actually fell out of popular usage for a while before the time when the Duck brand emerged.
This suggests that the phrase "duck tape" was effectively resurrected with a new meaning--that of a genericized trademark derived from the "Duck Tape" brand.
Google Ngram Viewer seems to partially back this up--note that "duck tape" declined in popularity from 1943 to 1963, slightly reemerged around when the Duck Tape brand got its name in 1975, but only really took off between 1987 and 2007: https://books.google.com/ngrams/graph?content=duck+tape&year...
I can't explain why usage increased slightly between 1963 and 1975, though. Hmm...
On a tangent to the topic of adhesive bandages, as a bit of a connoisseur of tapes myself, I want to mention that the best band aids I've ever found are Nexcare Waterproof Bandages[1]. Even if you don't care about waterproofing, they stay in place massively better than any other type of band aid I've tried. Years ago, Consumer Reports rated them as the very best as well. Most other adhesive bandages, especially cheap ones, are ridiculously awful, coming undone within minutes.
So the phrase "Duck tape" is fine, if you capitalize the first word and use it to refer to tape made by the Duck brand. Likewise, "Duck Tape" is fine, if you capitalize both words, since it is the name of the specific product line made by Duck.
In fact, if you call store-brand adhesive bandages "Band-Aids," and you call store-brand acetaminophen "Tylenol," you could argue that it makes sense to call store-brand duct tape "Duck Tape." And if you don't always capitalize "Band-Aids," you could argue that it makes sense not to capitalize "Duck Tape" either.
So I would argue that using the phrase "duck tape" to refer to duct tape is fine, provided you accept that it is effectively a genericized trademark.