Many of the squares and other bright spots are reflections in the glass window. Note how some are doubled (such as the pair in front of the chair). Note also the car headlight reflecting off the lower-right part of the picture.

The Radiomuseum learns that the cabinet is a Pooley Model 2000-R-2 "Arm Chair". It contained a powerful Pooley Amplifying Horn with AK driver, space for both "A" and "B" batteries and a charger, if desired. Beside this, there is a small drawer for tools, log card, etc.

In the Shorpy submission Atwater Kent: 1927 Mr. Atwater Kent himself was pictured standing by a Model 33 in a Pooley 1700-R-2 cabinet. the comments will give you a lot more information.

Lying on the chair, one of the most popular accessories for the well-dressed flapper in the mid-1920's. Memorialized in a song titled, you guessed it, Spanish Shawl!

Strange to see anything in a prominent window display that looks like a large family after 12 years have just given their tattered belongings back to the store for consignment.

The cabinet is an add-on feature: https://www.shorpy.com/node/4179

I think the backdrop is a rug because fringe is visible along the right side of the picture and the body of the design has a visible texture. Electric sound amplification was very new in 1926 and the sloping sound board was probably an Atwater Kent attempt to improve the broadcast volume. The right angle controls were likely to make chairside tuning easier.

I believe the hollow end of the radio cabinet is the horn for the speaker. So cool in my opinion.

... definitely looks like a rug. The two little squares indeed appear to be a reflection to me. Look to the center left, you'll see two similar reflections. The hollowed out open part is what I assume to be a feature to improve acoustics.

Isn't that speaker facing out toward the left? Sitting in the chair it would be radiating out to the window, wouldn't it?

Lots of Atwater Kent info here, but I haven't found that cabinet yet:
http://www.atwaterkentradio.com/index.htm

I love the history and quaintness of old radios but, like old cars, have no wish to own any!

First, is that busy busy busy backdrop actually a massive rug? Second, what are the two little squares on the chair? My instinct is to say a reflection but somehow that makes no sense to my eyes. Third, why does it look as though the radio's cabinet is hollow and open on one end, with an up-sloping part ... sorry I don't know how else to describe it. No I have not been imbibing unless you count sparkling water.