"Do you remember this kanji?"
If you think about it, it's the English phrasing that is illogical. Think about when you say "I go to school" versus "I am going to school." When you say the former, you don't necessarily that you are on your way. You mean that you do it on a regular basis, but not necessarily at this moment in time. In the second, you are saying that.
Now, when you say you remember kanji, do you mean to say that you remember kanji on weekdays but not on Saturday, Sunday, or holidays? Of course not! So in actual fact, you are remembering kanji. The present tense in English usage is inconsistent and illogical. Remembering is a continuous action.
I think that's a good way of remembering the distinction, but there are exceptions even in Japanese. For example, 知っている is usually translated as "know" in English (though it is more like "knowing" from a grammatical point of view as you say), yet 知りません is generally used for "not know" rather than 知っていない . I can't think of other exceptions right now, but there may be some.
It's also not as illogical in English as it may first seem because "know" is a stative verb (that is it expresses a state rather than an action) rather than a "dynamic" (action) verb and so our grammar makes a distinction with progressive aspect while Japanese does not. The wikipedia article does a good job explaining stative verbs https://en.wikipedia.org/wiki/Stative_verb. See especially the progressive aspect part.
Three months later it's still not accepted, even though it's the most standard way to write Duo's own sentence with kanji.
If a person was to type "konokanjiwooboeteimasuka" and then press space bar, この漢字を覚えていますか is almost certainly how every IME will convert this by default. Baffles me why they don't let more people contribute to this course to get basic stuff like this sorted out quicker.
I'm guessing that it is better to use 覚えています rather than 覚えます because the first form implies that you are remembering (continuously), whereas the latter implies a one time occurrence. I'm speculating based on use of 知る, where "Aを知っています" is used for the English equivalent "I know A" sentence .