Jetze Mellema, out of interest, would you mind sharing which https://docs.microsoft.com/en-us/microsoft-365/admin/manage/release-options-in-office-365 was set for the user who could access the "new experience" on your "personal tenant"? I appreciate the Office 365 release may not have an impact on Teams functionality (it's not specifically listed unless Teams is part of "Skype for Business and related services"):
The Microsoft 365 updates described in this article apply to Microsoft 365, SharePoint Online, and Exchange Online. They do not apply to Skype for Business and related services.
I also appreciate the feature may not roll out to all "Targeted release" tenants/users at the same time, but wondered whether the Office 365 release might be a clue.
Anyhow, on our work tenant I've now been added into the "Targeted release for selected users" list (could take 24 hours for changes to take effect).