Is the Unted States the only country to use income ltrs from them Embassy as proof of income?

All embassies do, but the US embassy is one of the few that doesn't require proof as one makes an affadivid.

It is the requirement by immigration for anyone using income for retirement to obtain such a letter from there Embassy.

A re-phrasing of the previous two replies might be that:

You are making a sworn statement as to claim of your monthly income before a US Consular Official but neither that official nor the Embassy itself makes any statement or takes any position as to the validity of such claim ... only that the person as identified by US Passport on said date appeared before the official on said date and made such claim.