Ian, according to Canon Omiya was only a marketing agent. If Canon propaganda is to be believed, the relationship with Omiya was simple. Canon made cameras, Omiya marketed them. Omiya made Hansa brand enlargers and other photographic equipment before Canon's founding and continued to do so until fairly recently.
I've seen Hansa enlarger lenses. Cheapies.
As for names, there were several. Precision Optical Instruments Laboratory was the first name of the firm we now know as Canon. Then Japan Precision Optical Instruments Laboratory. Then Precision Optical Industry Co. Ltd. And then Canon Camera Co. Inc.
See
http://global.canon/en/c-museum/history/story01.html for Canon's view of their history.