What does " under what name" mean in this context?

Question

I was in a foreign country wanting to head to the airport, so I called the taxi company using the hostel's phone and asked them if they could send a taxi.The lady who answered the phone asked for my address and phone number, so I responded by saying that I don't have a number in this country. After that, she asked me to give her the phone number through which I contacted them (which is the hostel's number). But the next question she asked baffles me. She asked, "Under what name?"

I wasn't sure if she was asking what my name is or the name of the hostel?
Therefore, I asked her what that means, but she refuses to clarify.
Afterwards, I gave her the name of the hostel , but she sighed and hung up on me.

Of course, a taxi was not sent to my place. So, what does she mean by "under what name" ?

Answer 1

Since you gave the telephone number of the hostel (a hotel-type of accomodation, with multiple guests), she needed to know the name of the guest the taxi driver should ask for.

"Under what name" is the usual way to ask how the hostel/hotel knows you - what name did you give to the hostel management. (Don't get stuck on the word "under" - you can substitute the word "as" for "under" - for example, "What name did you register as " -- or, as she said it, "As what name did you register?")

In the past, it sometimes happened that people would register with a hotel/hostel under made-up names, like Smith or Jones, if they did not wish to give their real names. This is still very common with celebrities - for example, Johnny Depp famously uses "Mr Stench", and Ozzy Osbourne is reputed to use the name "Harry Bollocks".

But the question "under what name" is a common question that someone would ask in order to find out how you are known to the hotelier where you are staying.