Yes, it is possible, but you will have a lot of work to do it.
First thing first, you need a way to communicate with your users. Drupal sends quite a lot of data to users, like password restoration mails. If you want phone numbers to replace mail, you need to be able to send SMS messages. SMS Framework seems to be your friend here. There is no stable version for Drupal 7, but as far as I know it is most stable at the moment anyway.
When you have a way to communicate using their mobile numbers, you need to override mail sending, to convert mails into SMS messages. There are, generally, two ways to achieve this:
SMTP Authentication Support - It is possible to configure mailserver to send sms messages if address is in firstname.lastname@example.org or similar format. This does not even require any SMS framework in Drupal, really.
Use Mail System and create your own
MailSystemInterface implementation that will detect sms addresses and use SMS Framework instead of whatever mail system you are using.
If you don't want to mess with mail systems, you will need to effectively write user module from scratch and replace core one with your own - it relies on mails too much, writing
alters would not be feasible.
Last thing: For all Drupal forms, you need to use form alter and detect required email, changing it into two fields, at least one required. And provide relevant storage. Actually, this seems to be an easy part.
Note pretty much every modern phone can receive emails, so I hardly see a point in this.