Duly is the adverb corresponding to the adjective due. The relevant senses of due here are appropriate, deserved, and justified. So when we say something is duly noted, we mean that noting that thing (i.e., noticing or acknowledging it) is appropriate and justified.
In official contexts, duly noted implies that the thing being noted will get appropriate consideration when the time comes. In less formal contexts, the phrase is sometimes used ironically, often to dismiss something that might be true but is irrelevant.
The phrase is rarely misspelled duelly noted because duelly is not a dictionary-recognized word and spell check catches it. But it is sometimes misspelled dually noted. Dually means twice, double, or in two different ways, so dually noted literally means noted twice or noted in two ways. It’s possible to imagine situations where dually noted would make sense, but the phrase is usually just a misspelling of duly noted.
The phrase is spelled correctly and used well in the following examples:
But the building is in a historic district, and the Landmarks Preservation Commission duly noted that the change had taken place without a permit. [New York Times]
They have a nice variety of gluten-free items available, duly noted on their menu. [Kennebec Journal]
All duly noted, but speed dating is a bit of a fad, isn’t it? [Financial Times]
Again Ilitch told Richner and Newman that their concerns were duly noted. [Ann Arbor Chronicle]
The creeping influence of Korea’s Pop industry has been duly noted by journalists, fans, and artists. [SSG Music]