Maya's bounding boxes are axis-aligned, but note that they are aligned to the node's local axes.
Every piece of geometry in Maya consists of a shape node plus one or more transform nodes in a hierarchy above the shape. When you "transform" a shape, by moving, scaling or rotating it, you are really changing its parent transform (or grandparent, etc), not the shape itself.
So if you look at the shape's bounding box, it will be unchanged because the shape is unchanged. Instead, you have to look at the bounding box of the transform node above it which was actually changed.
If you were to select a specific vertex on the shape and move that then you will see a change in the shape's bounding box because now you have changed the actual geometry which comprises the shape.
When you do a freeze transforms, that takes all the transformations in the hierarchy from the selected transform node on down and bakes them onto the object's geometry -- that is, it resets the local space positions of the shape's vertices to match their positions in the selected transform's space (which is worldspace if it's a top-level transform), then resets the transformations of the affected transform nodes to their defaults (i.e. no translation, rotation or scaling). That's why you see a change in the shape's bounding box, because the positions of the shape's vertices have actually changed.
As for oriented bboxes, you can sort of achieve that by grouping your object under another transform, then rotating that upper transform into the desired orientation. You can then take the corners of the object's bbox (in local space) and transform those points into worldspace to get an oriented bbox.