Since the upgrade of the License Manager from version 10.2 to 10.2.2 is a minor version upgrade, the labeling of the version does not reflect the upgrade.
To verify that the upgrade has been incorporated, follow the steps below.
1. Open the Control Panel > Programs and Features.
2. Find the listing for 'ArcGIS 10.2.2 License Manager.' The version number is shown in the Version column: 10.2.3552.