[wim]

Check and see if both skewer springs are on correctly. The small end of the spring goes against the axle ends. If the springs are reversed, the large end of the spring sometimes wraps itself around the axle and prevents the axle from going into the dropout. If you don't notice this common screw-up, you may think the dropout is too small when in fact, it's not.

 

[wim]

The problem is that the derailleur hanger slot doesn't line up with the frame's dropout slot. The silver derailleur hanger needs to be rotated counter-clockwise a few degrees so its slot can line up with the frame's dropout slot. But it looks like a gob of paint or some sloppy machining might keep you from doing that. Nevertheless, loosen the bolt that holds the hanger on just a bit and try to rotate the hanger counter-clockwise so the slots are lined up. Then tighten the bolt again.

If you can't line up the slots, it's time to scrape some paint or carefully file away a bit of material.

 

[wim]

If you look from the other side of the frame (NDS) and see the reverse effect (extra hanger material), why not file a little of the hanger away instead of filing the dropout?

Agree, that's what I would do if the hanger can't be moved. Sloppy workmanship, to be sure. But these Giordanos are marketed by Kent International of GMC Denali fame. So you can't expect perfection.

 

[wim]

....is this really the same frame as the GMC Denali and CCM Presto and Avenue, & Supercycle Medalist and Tempo? They sell tens of thousands of those every year for many years, they couldn't have all the parts that come off the tools like that.\

It's not the same frame, but consider that this misalignment issue didn't make itself known until the OP tried to put a quick-release axle into the dropouts. Point being: this bike came with a solid axle and axle nuts, which tells you something about its provenance.

 

[wim]

I'm still a bit surprised that the old wheel fit, but the new one did not as I thought the axle sizes were generally standardized.

Not across the board. Department store bikes often come with 3/8" diameter axles, which is just a tad smaller than the 10 mm rear axle standard.