Hello @dheeraj,
Unfortunately, this can be a tricky issue to troubleshoot as I believe we check for Esri licenses/libraries via the registry. I do have some preliminary steps that may provide us with more information.
1) This article provides ideas for you to try:
https://knowledge.safe.com/articles/315/geodatabase-formats-missing-greyed-out-or-have-unm.html
2) Are there or have there been any other Esri installations on that machine? ArcGIS Desktop or ArcGIS Server?
3) ArcGIS Pro is licensed on the computer correct? I noticed from your screenshot that you were signed into ArcGIS Pro, I just want to verify that it is licensed.
4) When ArcGIS Pro was installed, was it installed using 'Me only' instead of "all users"? If so, try re-installing using "all users".
5) Do you have the ArcGIS Pro Data Interoperability extension installed? If so, could you try running it and seeing if you can access the File Geodatabase formats(you should be able to with opening the Data Interop through ArcGIS Pro).
If you are able to use the File Geodatabase Open API format while we troubleshoot this issue, it may allow you to proceed with your work. As a note, the Open API format does have reduced functionality as opposed to the File Geodb format. More information regarding the Open API format can be found here:
http://docs.safe.com/fme/html/FME_Desktop_Documentation/FME_ReadersWriters/filegdb/filegdb.htm?Highlight=geodatabase" target="_blank">http://docs.safe.com/fme/html/FME_Desktop_Documentation/FME_ReadersWriters/filegdb/filegdb.htm
http://docs.safe.com/fme/html/FME_Desktop_Documentation/FME_ReadersWriters/filegdb/filegdb.htm?Highlight=geodatabase
