This is the expected behavior when you try to install the extension in Azure DevOps Server 2019. You do not need to install extra Test Manger for Azure DevOps Server 2019.
The official link is also clearly: This extension does not support the version of the Server you are currently using. See Works With for supported Server versions.
It only work Works with
- Team Foundation Server 2017
- Team Foundation Server 2018
Note:
- Manual testers do not need this extension and can execute tests as a
Basic user in an Azure DevOps organization/collection. Read more on
this here.
- This extension is available out-of-the-box in Azure DevOps Services and Azure DevOps Server (2019 onwards) and hence doesn't need to be acquired for them.
We strongly recommend you to use Azure Test Plans or the Test hub in Azure DevOps Service/Server (a fully featured Test management solution) over Microsoft Test Manager for all your test management requirements. There will be no new versions of Microsoft Test Manager.Microsoft Test Manager 2017 (which shipped with Microsoft Visual Studio 2017) is the last version.
More details please take a look at our official tutorial here.
Besides, to use Azure Test Plans in Azure DevOps service you need some License requirements
Azure Test Plans uses an access level called Basic + Test Plans, which is available from Azure DevOps.

For the Manual testing permissions and access suggest refer this link: https://learn.microsoft.com/en-us/azure/devops/test/manual-test-permissions?view=azure-devops#license-requirements
Hope this helps.