Backup runs in the context of the account running SQL Server - and since that account is local system it does not have access to the UNC path. It doesn't matter who is logged into SSMS or how you are logged into SSMS (Windows or SQL authentication) - the command still runs in the context of the account that is running the SQL Server service.
You should also be aware that access is now setup/granted using the per-service SID accounts (NT Service\MSSQL$SQLSERVER). Since you cannot grant access to a UNC path to this account - you need to be able to identify a domain account that will be used.
The simplest method is to run SQL Server with a dedicated domain service account - then you can grant access to the domain account for those network resources. The other option is to create a proxy account and then run the job from SQL Server Agent using the proxy account.
Using a proxy account will not allow access to the UNC path from SSMS though...it will only work through SQL Server Agent jobs.
Here is a document on how to grant access to the per-service SID: https://docs.microsoft.com/en-us/sql/database-engine/configure-windows/configure-file-system-permissions-for-database-engine-access