Did they use those exact words? Sounds strange to me as most American English speakers would say “phone number” or “cell phone number”, and both are used pretty much synonymously (because in practice they basically are) for a long time. More formal forms (government forms) might ask for something like “daytime phone number”. In particular I haven’t seen the word “telephone” in ages.
They weren't asking for home number, work number, and mobile number. Or as you mentioned: daytime number, evening number. It was more along the lines of: Name, Address, Mobile Number, Company Name, Job Title, etc. They specifically wanted a mobile number and no other type of number.
Were they wanting a mobile number so they could more easily research an applicant using their internal tools? Only they would know. But it drew my attention to the possibilities and the importance of obtaining a mobile number.
Back to OP's question, part of the reason a mobile number is required to create a Facebook account may be to help tie different types of records together. Of course, as mentioned, there are other reasons, such as security. Users will understand and provide a mobile number for that reason alone.
In business settings it’s still pretty common to ask for a cell though, or so I thought.