The mixture is largely water so yes it does saturate into the wood; the lime is dust so even subsequent coats penetrate. Also, the more coats the whiter it become; the 1st coat goes on translucent, leaves one wondering if a mistake has been made, drys to white.
There is no change to the surface of the wood, does not harden.
I too have read it is anti-fungal.
As I indicated above this is back in the day stuff. Not a lot of science just decades of farmers using it inside barns, coops, etc. and word of mouth collaborating it works.
I bought a bag of hydrated lime, did two coats and still have half the bag left. $17 for a 6kg bag at the time I purchased, $23.99 today.
Hard to understand why one would not use it for animal enclosures if other than a wood finish is desired.