How to rsync files with matching pattern in path by keeping directory structure intact?

I guess you're looking for this:

rsync -a -m --include='**/commonname/*.foo' --include='*/' --exclude='*' root@1.2.3.4:/var/lib/data /var/lib/data

There are 2 differences with your command:

The most important one is --include='*/'. Without this, as you specified --exclude='*', rsync will never enter the subdirectories, since everything is excluded. With --include='*/', the subdirectories are not excluded anymore, so rsync can happily recurse.
The least important one is -m: this prunes the empty directories. Without this, you'd also get the (empty) subdirectory /var/lib/data/sub3/sub4/differentname/ copied.

CodeHunter