Is it possible to increase only the vertical size of a shape in PowerPoint?

With the object you want to "stretch vertically" selected, click the "add custom animation" button. Choose "emphasis," then pick the effect named "Grow/Shrink." You may have to expand the list of "emphasis" animations to see the "Grow/Shrink" effect. Within the custom animation window, as one of the selectable properties for this effect, there is a drop-down for "size." When you expand that drop down, you should also see three options at the bottom of the size presets to specify if you want the shape to grow horizontally, vertically, or both (both is the default).

I've used this to make an object appear to rotate around a horizontal line by setting the size to 0% and choosing vertical only.

No there's not, but you can approximate this visual using two objects

I don't think there's a native way to do what you're looking for, but this work-around might give you the same visual effect. This technique relies on creating two shapes: one for the starting size and one for the ending size, and then animating the end shape. Also, it may look better with a squared edged box, although I know you're using a rounded one.

1. Create a box that will be your starting size, set the fill color as appropriate and turn off the border.
2. Create a second box at the right size and location for the end state, set the fill and border the same as object #1, and layer it behind object #1.
3. Set the following animation parameters on the 2nd object:
   • Set the entrance effect to "split"
   • Set the direction to either "horizontal out" or "vertical out" (which will depend on the motion you want and the object orientation)

When you preview the animation, you should see object #1 in the foreground and object #2 growing out from behind it straight up and down.