Equivalent of Shape Merge in Blender?

Is there the tool that would do the following?

You have a cube mesh, for example, then a circle shape is drawn on top of one of the faces.
You can then create a compound mesh, with a circle shape superimposed on the cube.
You could then extrude the circle shape mesh out of the cube.

This link might help as an example:

http://docs.autodesk.com/3DSMAX/15/ENU/3ds-Max-Help/index.html?url=files/GUID-50D6A2B6-D4D

Not sure what relevance your link has

Knife Project ? http://wiki.blender.org/index.php/Doc:2.6/Manual/Modeling/Meshes/Editing/Subdividing/Knife_Subdivide#Knife_Project

The link DOES not take you to the shape merge I had picked!!! shame on you Autodesk!
I see what you mean about the link, but you got the idea of what I wanted…

Is there the ability to cut the mesh with the knife by selecting a shape such as a circle rather than a freehand drawn shape?

You can boolean modify the cube to be cut by the circle. It will make some messy geometry, but just clean that up and you have a nice circle hole in your cube.

Don’t use booleans for this. Knife Project works just fine. You can use circles, squares, freehand outlines, whatever you like.