A few points here, at least starting w/ My Opinion:
I would imagine rather than a private label like Sears which the retailer would be promoted more than a "Coldspot" or a "Kenmore", that we're talking Roper for the range and Whirlpool for the fridge (though the in-laws had a top freezer Coldspot that those handles on the refrigerator remind me of, as well as other Coldspots I've seen)...
Then, seeing 'Vivian Vance's Fridge' being the topic, it to me, meant non-other than Cheap Fred only springing for a Monitor Top that at least uses electricity, as opposed to ice, which I saw on at least one episode of I LOVE LUCY...
Then, the Swan brand, I no longer see for any kind of detergent or dish soap (and I probably could only remember seeing it on the sink top only vaguely!) but now-a-days, for Cake Flour, and with a very similar logo...!
-- Dave