More recent scholarship has found that his armies were really much smaller.
Basically, in every country where the Mongols showed up and kicked ass, their historians explained it away by claiming to be outnumbered - hence the term "horde" becoming synomous with massive numbers.
In reality, the Mongols were pretty badly outnumbered in pretty much every battle they ever fought, and won through superior discipline and tactical skill.
The 300,000 figure was total forces spread out from China to the Caucuses in many different conflicts. My impression of he term “horde” is that they were so mobile they could appear 100 miles away a day later which made their numbers appear much larger than they actually were.