What would a bracket for holding 1 5/8" square post be called?
Rather than use with brackets why don't you consider the use of some 3/8" x 2.5" lag bolts installed through 7/16" holes drilled through the post. If you do this remember that for best grip of the lag bolts into the surface behind will be to drill the proper sized pilot hole. In this case the pilot drill can be 1/4" size.