There's no way it fits any of my 1/2" stuff, but it fits a vintage Snap-on ratchet that I bought just fine.
Interchangeability from mid 20's to modern, same drive size, can be a little sketchy, not just Snap-on, but it has been observed before with them, too, due to manufacturing quality.
The socket is marked Snap-on 1 1/8" S360 1/2 (1/2? what kind of witchcraft is that?)
Take a breath, maybe a take a drink, and put your thinking cap on because this will get a little nutty to read!
The "1/2" marking is a suffix to the part number. In the mid-20's, when they were making the 5/8-inch drive (dubbed "Heavy Duty") that
@snapmom alluded to, they were making 1/2-drive (dubbed "Master") at the same time. The problem is they didn't foresee the overlapping range (three largest openings in Master drive range being the same sizes as the three smallest openings in the HD drive range) issue coming when they - believe it or not - re-used the same part numbering scheme for the HD drive sockets. As a fix, they added a "1/2" suffix to the biggest 1/2-inch drive part numbers in their range to distinguish them from their 5/8-drive near-twins. At the time, it was only 300 1/2 (15/16"), 310 1/2 (31/32"), and 320 1/2 (1"), whereas 300, 310, and 320 (no suffixes!), were 15/16", 31/32" and 1" respectively in HD or 5/8-inch drive! (I warned you it was nutty!) But then, when they started making 1/2-inch drive sockets with larger openings, they kept the 1/2 suffix scheme going! Eventually they dropped the 5/8-drive, standardized on 3/4-drive, and gave them completely different prefixes. But those "1/2" suffixes prevailed.
By P/N, your socket should be a standard 1/2-drive socket with a 1-1/8" service opening.
The "E" could be a date code (1944).