I just had this exact same issue when updating to the most recent 5.0.6.7. The thing is when the updater starts it shows a message "stopping the service". The service was stopped by the installer but I saw two "BlueIris.exe" (but not the service) processes as suspended in Task Manager. I recently updated to the latest 1909 build of Win10. I wonder if they changed something, so instead of stopping processes upon installer's request they get suspended instead. Or maybe Ken is using a deprecated function to stop them. I will see how the next update will go.
Killing the suspended processes allowed the installer to finish.