It's a weaving device and may have a specific name. Even though I've had one in hand, I am unable to find an appropriately specific term.
The threads for weaving (often for lace) are wrapped in a specific pattern on the pins and the crossing threads are woven in alternating patterns along the radial threads.
Once the pattern is completed, the pins are retracted, allowing the work to be removed. Many of these completed items are then stitched into a larger piece, such as doily or tablecloth.
Our makerspace has had a few weavers pass through and this device is common among the varied weavers.
Apparently these devices are called "daisy wheels" or "daisy knitters". They were a novelty in the 1930s (usually made of metal) and had a short revival in the 1970s and 80s (usually made of plastic). The plastic variety was sold under the brand name "knit wit" and another brand name for the metal tools was "crazy daisy winder".
There are similar (modern plastic) tools that have removable pegs instead of retractable spokes that are called "flower looms" or "bloom loom".
I stumbled upon the identification of the tool and an example of what to do with it in this Youtube video by Retro Claude.
Here's a vintage daisy knitter with the original instruction manual with some example patterns provided by the South Canterbury Museum:
You're supposed to create a number of individual daisies with it and then connect them by sewing or crocheting the edged together to form an object like a cardigan or afghan.
Here's a vintage PDF pattern on etsy for a sweater that includes these daisies:
The actual techniqe and idea seems to be much older and possibly originated in / around Teneriffe. Here's a link to a free 1904 book that explains many different styles of Teneriffe lace.