Hi,
As of writing this, it's currently not possible to have multiple instances of Microsoft Teams client running.
A request was to be able to have multiple chat windows open at the same time, and this feature was implemented by allowing pop-out chat windows (although this isn't what you're asking for).
Your query is also discussed over here:
https://medium.com/@seanwallbridge/how-to-have-multiple-microsoft-teams-instances-windows-e802244ecb98
You can submit feedback about this over at the Microsoft Teams uservoice over here:
https://microsoftteams.uservoice.com/
There might already be such suggestions, so you can search first if there are any.
(Please don't forget to accept helpful replies as answer)
Best regards,
Leon