You don't need to use a character class around <, and you should probably not use a greedy .* match (would break on <<Label1>> <<Label1>>).
This worked for me:
syntax match PLS_label "<<[^>]\+>>"
(though you only get colors if you use a highlight command on PLS_label).
OP points out a region using << ...
My problem was I was trying to source my vimrc file within gvim/vim and my changes were not being applied in the session. I would have to reopen gvim/vim to see the changes. Adding the $HOME user environment variable: C:\Users\<username> resolved this.
For my setup I am using Windows 10 and vim/gvim 8.2.
Instead of using the user vimrc file: $HOME\...