I have a customer that is updating their license manager daemon from 11.13.1 to version 11.16 but the latest license daemon download on shown here https://www.intel.com/content/www/us/en/programmable/support/support-resources/download/licensing/da... is 11.13.1
From where is your customer updating the license manager? Intels tools support 11.13.x . If its for another tool, you may want to install it in a separate path.
In general , the newer license manager should also support the old license files. Shutdown the existing license server / daemon, install the latest one, restart the new license server/daemon with the license file. It should pick up the existing Intel tools license files and run. Then launch the respective tools and check if the license is being checked out and the tools are working. If so, then you have no issues to worry about. Else , post back here with the details of the License diagnostics.