Does this mean that you can drive in the forest for an entire day and still be in the forest?

Probably, yes. This is possible in Sweden, if you go from South to North, you can travel for multiple days by car and if you avoid highways, you will not leave the forest at all.

The joke I'm making is that many Texans like to make that statement about Texas (with regards to size and driving) and claim it's unique to that state and to the US without realizing that it's common for many other parts in the world as well.

Pretty much. Try driving from Perth to Broome in a day. It is about 24hours of straight driving and it is still a good 10 hours to the boarder.

Fun fact is: I heard Texans saying it only a few weeks back. Now I get what you meant ;)

You can do that in Virginia if you drive slowly enough, stay off the Interstate.

For example if you go from Cumberland Gap to Virginia Beach, a distance of 499 miles, it will take you 10 hours and 25 minutes.

Well if you drive slowly enough, you can do that in my backyard :D

This is common in Brazil.