The SERVICE in running on a remote server - and SSMS talks to the service.

So of course the disks available are SERVER-CENTRIC.

You could create a share or browse to a url that is on your local machine - such as if your workstation was called \\MYWORKSTATION you would browse to \\MYWORKSTATION\C$ - that's if C$ was an admin share available to the SERVER.

But more important - why do you want to do this??

If this is a production server - then you want regularly scheduled backups to occur on only local disks to the server

If this is a one-off backup then open a share on your workstation that the server can see.

I've done commands like this to send backups to remote folders

Code:
Backup Database TCSHemp To Disk='\\10.14.14.98\tcs folders\ARCHIVE TCS\TCSHemp.F07
Remember that the backup runs under the USERNAME of the SERVICE running on the remove server - not under your username on the workstation. That's only how you authenticate to connect to the SQL server "service".