What’s the “Visa” word connection? There’s the “visa” brand of credit card, but certain types of passports are called “visas.”
Visa is a branding exercise for a particular credit card company. It says to consumers, “Take me! I’m your entry into a world of consumer goodness and happiness!”. It is analogous to the original travel related visa which is NOT a type of passport. Rather, a visa is a permission document (time/activity/location limited) for entry to a country you are not a citizen or resident of.
“Visa” in terms of passports - you are allowed access into that country.
“Visa” in terms of the credit card - you are allowed access to more stuff through credit.
It’s a play on words type of thing.
Clear as mud, now? 
A visa isn’t a passport as such - it’s an endorsement in a passport (a stamp, barcode or sticker), or a separate document used in conjunction with a passport giving permission to enter a country. It shows the immigration control that you’ve been pre-checked. You only need a visa if you’re going somewhere where the government doesn’t like your government, or thinks you’re a security risk or an illegal immigrant risk, or they just demand you get one to make money.
So, as previously stated, it’s a neat brand name. There’s a UK credit card called “Access” for the same reason.
“Visa” was originally “BankAmericard” (just as “MasterCard” was originally “Master Charge”). The Bank of America started out marketing their own credit card to other banks, but the division was spun of, and eventually they decided they didn’t want to keep the connection to the parent company. So they changed the name. They picked “visa” because of its connotation of world travel, and because the word is pronounced the same way in most languages.
So this means I won’t have state department immunity if I run over someone in a Dodge Diplomat?
IIRC Honda marketed a scooter called the passport and made a commercial to make a similar connection. Guy pulls up in a car to the curb in front of a restaurant and the Maitre D’ comes out shouting in an exxagerated French accent, “you cannot park here without a passport” then someone in a scooter pulls up and is welcomed.