On those occassions when my DH has gone a wee bit overboard on the congee making, we have frozen it. No particular changes in texture as far as I recall. We tend to make plain congee. You might have a difference if you did chicken or maybe pork with preserved veg. Like any frozen thing, I wouldn't let it hang out too long.
The other thing we have done is cooked it part way and then frozen it. Then when you want to eat it, you can cook it the rest of the way--15-20 minutes or so.
But honestly, we don't have much spare congee hanging around. The pups are always up for congee in the lunchbox [and with 3 of them now, that just doesn't make for much leftovers!].