If you want milliseconds instead of nanoseconds precision, you might use
%3N instead of
$ date +'%d/%m/%Y %H:%M:%S:%3N'
or with your desired use of
$ echo "$(date +'%d/%m/%Y %H:%M:%S:%3N')"
But be aware, that
%N may not implemented depending on your target system or bash version.
Tested on an embedded system »GNU bash, version 4.2.37(2)-release (arm-buildroot-linux-gnueabi)« there was no
date +"%F %T,%N"