Should I upgrade compatibility level of MS Report Server database
This is not a question we can decide for you. Please review this Microsoft document especially the section with heading Differences Between Lower Compatibility Levels and Levels 110 and 120. Go through each item and decide for yourself.

Once you test thoroughly with new compatibility level you will know if you have any reason to keep old compatibility level (for backward compatibility). I prefer to use new compatibility level as you can see newer versions of SQL Server only support up to 3 or 4(maximum) older versions.  If you do not do it now at some point you will be upgraded anyway.