Transforming Shapes like illustrator

Are you talking about Parametric shapes like the Rectangle or Bezier shapes (also mask paths)?

There is a very good reason that clicking and dragging on a shape in a shape layer or on a mask does not change the size by moving one edge. Clicking and dragging will not change the size of the shape, it changes the layer scale. If you want to change the size, you need to dig into the shape layer properties.

Dragging on a path shape in Illustrator does not change the scale, there is no automatic scale for vector paths in AI. If you use the Scale tool in AI it will work exactly the same as the scale works in After Effects, you'll have to move the "anchor point" to get the object to scale from one of the edges.

If you convert a parametric shape (Rectangle 1)  to a bezier path, select the path in the timeline so you are editing the path and not the layer, then double click or press Ctrl/Cmnd + t to edit the path you will get exactly the same behavior you get in Illustrator when you double click on a path and move the edges. The behavior is identical. You are just not digging far enough into the UI. You just have to make sure that the Path is selected in the timeline. When the path is selected in the timeline the selection tool (v) changes to a white pointer so you know what tools you are using. Most AE users fail to notice this difference. As long as the path is selected in the timeline, the tools work just like they do in AI. 

Layer transforms are always centered on the Anchor Point. That makes sense if you are intending to animate any of those properties. All the additional controls for shapes give you more tools to control the animation. Modifying the shapes in the timeline is extremely easy and makes animation easier than it would be if you just double-clicked a layer and changed the shape. A lot of people have a very difficult time editing paths and shape layers in AE, but if you understand the UI and pay attention to what the tools look like when they are selected or modified by Ctrl/Cmnd or Alt/Option, you should have no problems editing paths or polygons in AE.

There is a scipt called "move anchor point" which let you set the layer anchor to one of the sides. You can also use the pan behind tool and press ALT, or something to stick it to the edges.

On a shape, I always fix the shape anchor to one of the side using expressions to hold it and transform the other side.

I'm sure somebody already wrote a script for this - so have a look at google.

There is no way avoiding the anchor point. This is math behind this all. In other Adobe apps, this is not exposed so much to the user, but in AE it is.

*Martin