Why mongo write only one partition
There is no inconsistency here, this is simply due to how you have the volumes mounted and where you have located you MongoDB database path.  You have one logical volume mounted on root (/) and the other mounted on /home - I am guessing you do not have your database path in MongoDB set to something under /home instead it is set to something else which is under the root path.  

The most common database paths for MongoDB are /data/db (the default) and /var/lib/mongodb (default on Ubuntu for example), but you can set it to whatever you want.  Given how you have your mount points configured, both of those common defaults would go under the root mount point and hence onto the /dev/mapper/vg_db1-lv_root volume.  If you switch your database path to be under /home or you re-do your mount points appropriately the data will then be placed on the second volume.