The USPS is a government agency that already has physical branches in almost every community in the US, which would make it an extremely accessible bank for the 4.5% or so of households who currently don't have a bank.
It could also be pressure on American banks to treat their customers better, since the USPS is not burdened with a responsibility to generate dividends for shareholders.
The USPS is also not directly funded by taxes (though it is indirectly subsidized via it's tax status and sole right of access to mailboxes), and with the waning popularity of letter mail, would become more self sufficient in the long term if it branched out into other services
Other countries also combine banking and postal branches for similar reasons, like Japan (although it is in the process of being privatized).
I guess you haven't traveled much. Post offices in other countries also provide fundamental banking services. Right now, one can get money orders from USPS.
USPS isn't a company, it's a government department which provides a public service (deliver mail at very low prices). Because the USPS has so many branches all over the country, many people (myself included) think it would be a good idea to also offer banking services and so that the unbanked can become banked. Even just a basic checking account would mean people don't have to pay check-cashing fees anymore.
AFAICT/R, JP Post banking used to be really barebones, even for late 2010s Japan. No credit cards/debit cards, not a lot of financial products, etc. In recent years it seems like they have added a credit card and a debit card and online banking as well.
On the flipside, they are one of the most ubiquitous ATMs in Japan and your local post office will double as a branch for JP Post Bank (which are everywhere).
Oh and it's nation wide too! Which is kind of important for a country where some banks restrict activities and branches to their direct locality.
Man. I wish patio11 would talk about Japanese banking more because to most American audiences that system is also just, completely something else.
Huh? Why? Isn't the USPS a mail company? I don't get what that has to do with banking.