@Sedmák Jakub, Thanks for posting in Q&A. From your description, we understand that you manually update the product on the device, but Intune will still install the old version on your device while you have set “MSI product version check” operator ”Greater than or equal to”.
For our issue, I suggest check the MSI product version on the device with the updated app to double confirm the version is greater. To verify the updated version is greater than the old version, please run the script in the following link in Windows PowerShell ISE, here is the test on my device.
Getting version information from Windows MSI installer – JoeLi's TechLife (joelitechlife.ca)
If the updated app version is greater than the old version, but it is not detected, please open case to get more help:
Get support in the Microsoft Intune admin center - Microsoft Intune | Microsoft Learn
Please check the above information and if there’s anything unclear, feel free to let us know.
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