How do you obtain a short number from cell phone carriers to send text
messages to?  For example: 46645 is the number used to send messages
to Google mobile.

the short number is called a CSC or Common Short Code. unfortunately
the answer is probably not what you were hoping to hear.

In order to lease a CSC, an individual or company must first set up an
account on the CSCA website. Once this is done, an application for the
lease of a CSC may be completed. The application will then be either
approved or disapproved, and the applicant will be notified by email.

A CSC is leased for a period of three (3), six (6) or twelve (12)
months for a non refundable fee of $1,000 a month for a "Selected CSC"
codes and a nonrefundable fee of $500 per month for a "Random CSC".
The entire lease payment must be paid before the CSC is assigned to
the applicant and "registered".

you can create an account and application at:
https://www.usshortcodes.com/csc/AccountCreate.jsp