No it is not the default Android recovery. It is a custom recovery but it comes along and is part of the Open OS. Fortunately, since it is way more advanced than the default recovery (which comes with the FPOS). Particularly the possibility for making decent system backups is widely appreciated (though with the current TWRP version cannot handle encrypted /data partitions).
With the FPOS you need to flash it yourself if you want to have TWRP.
If, on the other hand, you want to have the default recovery for the Open OS (But why on earth would anyone want to do that??) you could try to extract the recovery.img from the latest FPOS and flash it via fastboot on your system. Though I am not sure if it is compatible or if it will work, but I guess so.