How are you spelling, or how do you spell?
Solution 1:
I'm reluctant to make assumptions here but I think that form is quite common in Indian English -- are you dealing with outsourced call centres?
The only alternative in which it's not wrong is along the lines of "how are you spelling your name this week" -- an unlikely question.
Solution 2:
This makes me think of a current trend on TV to say "I'm loving what you've done here." eg on fashion or makeover shows, expressing admiration for changes made.
If you google "I'm loving this" you get lots of hits such as this one from Cambridge.org
Most good grammar books advise students, “Don’t use stative verbs in the present progressive.” For example, one shouldn’t say, I am knowing the truth but rather I know the truth. The verb know is stative because it indicates a long-term condition, not a discrete action. The same holds true for ordinary uses of agree, hear, hate, love, think, and want and many other verbs.
However, your students may pose counter-cases. What about I’m agreeing with you or—even more colloquially—a fast-food ad that claims I’m loving it? First, let’s acknowledge that rules about verb usage are really more like guidelines. If discourse conditions require language to shift, it will shift. Few people would call foul on I’m agreeing with you. It means “I am saying something that shows agreement” and is clearly distinct from I agree with you (“In general or over the long term, I have an opinion similar to yours”).
In the fast-food ads I’m loving it draws attention to the speaker’s being in an enjoyable moment, not in a long-term state of enjoyment. The intended meaning is “I have my burger and this moment is great.” Similarly, if I whisper to a colleague at a meeting I’m hating this, I want to say that things are temporarily unpleasant, but this too will pass. In I’m thinking blue for that wall, I’m trying to say, “My current thoughts are that blue would be a good color, but I might think differently later.”
This agrees with your views on the temporary nature of How are you spelling that? Even though the spelling is not temporary, it seems to be one of the ways language use is changing.
Solution 3:
I think the the way they asked infers that they recognize the name, and respecting the possible uniqueness of its spelling (if any). I think asking "how do you spell your name" almost sounds like "i don't even know what you said, please spell it", almost as if it were asked by a clerk at the DMV.
Solution 4:
I imagine it might be more a matter of pride than anything else. The question "How do you spell your name?" suggests there is a generally knowable fact, i.e. the way your name is spelled, which the asker does not know. On the other hand, the form "How are you spelling your name?" implies that the spelling of your name is subject to your whim, and therefore the asker can be excused for not knowing.
Put another way, "How do you [do something]?" is the kind of question you'd expect an ignorant child would ask to someone who knows more about the world than they do, establishing the relationship between asker and askee as one similar in subjugation and humility to that of a student and teacher. On the other hand, "How are you (personally) [doing something (lately)]?" implies that the subject of the question is unknowable by any but the askee because it depends on their personal preference and history.
In short, the wording you're asking about might be used to preserve the asker's sense of maturity, intelligence, and pride because they may perceive the more common wording as self-demeaning, at least in this formal context.
I doubt they gave it this much thought though, it's just a plausible explanation for their instinctive use of this wording.
Or maybe it just sounds fancy and they like that.