Even though this looks like a limitation of Discourse because a multisite setup only allows you to use a single account for sending, this is in fact a limitation of your email provider where they only allow you to use a single From address under your account.
The fact that the SMTP_USER_NAME looks like an email address is because they have chosen to enforce that the From: header of the mails you send have to be the same (or: are rewritten to be the same) as the SMTP_USER_NAME.
The email headers are being set by Discourse and if you use an email provider that doesn’t touch them and allows you to use multiple from addresses under the same account, everything will work fine.