Do bees think it's summer underground?
Sort of.
According to The Don't-Starve Wiki:
Even though Caves have their own day and night cycle and are in a permanent summer, Bee Boxes built underground will not spawn Bees. This is true even if the Box is built in a place where natural light can be found. The same applies to the Ruins since it is considered a permanent night there. However, if the Boxes are far enough from the player, the simplified model of Honey production will take over and Honey will be produced at a rate of 1 per day, in both Caves and Ruins.