For running multiple MS Teams accounts side by side. Save this with the name of the MS Teams profile you wish to use. When launched, a folder will be created in your user profile. See https://danielsmon.com/2020/04/02/multiple-ms-teams-accounts-on-the-desktop/.

New MS Teams Profile.cmd

@ECHO OFF

REM Uses the file name as the profile name
SET MSTEAMS_PROFILE=%~n0
ECHO - Using profile "%MSTEAMS_PROFILE%"

SET "OLD_USERPROFILE=%USERPROFILE%"
SET "USERPROFILE=%LOCALAPPDATA%\Microsoft\Teams\CustomProfiles\%MSTEAMS_PROFILE%"

REM Ensure there is a downloads folder to avoid error described at
REM https://gist.github.com/DanielSmon/cc3fa072857f0272257a5fd451768c3a
mkdir "%LOCALAPPDATA%\Microsoft\Teams\CustomProfiles\%MSTEAMS_PROFILE%\Downloads"

REM Fixes missing "Desktop" folder
mkdir "%LOCALAPPDATA%\Microsoft\Teams\CustomProfiles\%MSTEAMS_PROFILE%\Desktop"

ECHO - Launching MS Teams with profile %MSTEAMS_PROFILE%
cd "%OLD_USERPROFILE%\AppData\Local\Microsoft\Teams"
"%OLD_USERPROFILE%\AppData\Local\Microsoft\Teams\Update.exe" --processStart "Teams.exe"

I am having the same issue. I have recently joined a project and I was invited to several channels for the new project in another organization, I don't see anything unless I switch the organization, but then I miss the notification from my own organization.

I have been surfing the web for a solution, but nothing works. It seems that the "CustomProfiles" aren't there anylong.

I played around with this again. Script still works for Teams Classic. The "New Teams" is a UWP app and I didn't find a way to open a second window.

As an alternative you can create multiple profiles in Firefox or Chrome and open a Teams instance in each browser instance