What is this?

Ice pick. Patented by William F. Arndt and John H. Ganzer as US Patent No. 2,070,773 on Feb. 16, 1937.

"This invention relates to ice picks and it has for its object to provide a simple and convenient device for removing ice cubes which have been formed on a cake of ice and disposing of them as desired."

That's an ice chipper! It's seems Coolerator refrigerators weren't cold enough to actually freeze ice but you could order a big block of ice and it would hold for awhile in the fridge. That tool is part of the process of getting ice cubes from the block. Collectors Weekly describes how the tool is used. For more context, this Youtube video shows the whole process of getting ice cubes from an ice block, but uses a slightly different Coolerator tool to break up the cubes.

A 2016 PDD post goes into more detail about Coolerator refrigerators.

This led me to a fascinating search for the true history of Coolerators. They were still powered by a big block of ice at least until the 1940s, as shown by this ad from the Duluth Herald from May of 1940. This is a great example of how terminology changed (or didn't change) in fairly recent history. There are still people alive who remember the ice man delivering ice for their ice box, which was also called a refrigerator, but didn't have coolant. And "making ice" meant chipping off from the big block. There's likely another post on the way.
 

Note in the ad above, it says they only ice their Coolerator every 4-7 days. That means a big block of ice was delivered from the Duluth ice and fuel company, which was promoting the use of these devices that needed ice. Little did they know, someday the need for purchasing ice would be gone.

I have a little projector made in the 1930s or '40s, by the Jam Handy company. It was a company (in Chicago, I think) that made promotional filmstrips. Inside the projector was storage for three rolled up filmstrips. One of them was selling GE refrigerators right at the time you mention, the transition away from the iceman. The writing and photography are something else. Such existential dread back then. Here's a sample, I need to make a slideshow or something for the whole thing. More than 70 slides to tell this story.
  

Wow, thanks Brian! That looks like definitely 1930s or maybe even late 1920s. The electric refrigerators were really expensive and had some dangerous chemicals so they didn't really catch on until the 1950s. Another story goes that women were getting too friendly with the local ice man while their husbands were away at war, so men were motivated to get electric refrigerators after WWII because then there was no need for those weekly visits. This site has some milestones (although some are inaccurate as far as decades or understanding the use of the word refrigerator).