Only if this his standard govt issued phone. It's also been shown they are also using their own personal phones. The could easily be using unapproved phones some random DOGE'er bought gave them with an MDM setup, without any real oversight.
No. Even if you managed to get the app and push it to devices, you can't just use TM-SGNL without having an archiving account from Telemessage.
Source: I manage this exact setup for several clients.
Why wouldn't the government (DOGE in this scenario) be able to get an archiving account?
edit: found their install doc! https://smarsh.my.salesforce.com/sfc/p/#30000001FgxH/a/Pb000...