From what I can find even in extreme case it's not true. Even with a fridge from 1972 which averaged 1726kw/h (Source: https://www.annualreviews.org/doi/10.1146/annurev.energy.27....) to a more recent one which will probably be around 400kwh. And with the kwh at 0.15$ (https://fred.stlouisfed.org/series/APU000072610).
You obtain a reduction of (1726-400) * 0.15 ~= 200$ a year.
I believe the average refrigerator price in the USA is over 400$ so I can't understand where the 2 years figure would come from without even talking about trying to supposedly be "green" ... And after 1988 you are down to 800kwh so you would gain something like 60$ a year.
Once you account for manufacturing and disposal, running the older is probably almost always better.
From what I understand high end fridge are often less efficient because of features like ice fountain or freezer in the lower part with drawers instead of doors.
Anyway I'm thinking it's probably not just post-facto justification because it looks too much like a seller pitch and wondering where it comes from. I'm French and I had never heard it.
2. Other countries have smaller houses, therefore smaller kitchens therefore smaller refrigerators. Between that and higher energy efficiency standards in general, new refrigerators in my local store vary from 230kwh (A energy rating) to 260kwh (F energy rating) in power usage.
So even when doubling electricity price I fail to see how it could be possible to reach the two years originally mentioned.
And I believe each time I have seen this 2 years figure mentioned it was probably USA centric conversation. I'm French and I had never heard of it before.
Inevitably the people making these claims sell HVAC systems.