Verizon and Wells Fargo requires unsecured networks for account updates on iPhone

The managed Wi-Fi networks are cellular offload networks from your carrier. They do not indicate the iPhone is managed, nor is any of this particularly related to add-on VPN apps. Your carrier wants you off their cellular network, when feasible.

As for web browsing and app network connections and such, that’s defaulted to secured end-to-end encryption for a decade or so. That’s what HTTPS is — end-to-end encryption.

I’m unclear why Verizon or Wells Fargo are involved in this discussion, except possibly by providing (open?) Wi-Fi networks themselves.

Do you have a particular error or issue you’re encountering?

As MrHoffman writes, the managed WiFi accounts are profiles loaded on your phone by your network providers to allow you to use their public WiFi when out and about. There's nothing sinister about them.

As to security - why do you think they are not secure? If you don't need a password it's because the profiles take care of that. However, if you are concerned about their security then you could install a VPN but be mindful of the fact that many VPNs, especially free ones, pose more of a security risk than the WiFi networks you are trying to protect yourself from.

Note that Wi-Fi security only protects the "local hop" between your device and your Wi-Fi access point. (Then, what sort of protection you get depends on the particular security protocols and devices in use.)

Wi-Fi security does nothing to protect packets once they leave your home, office, etc. and go over the Internet.

Visiting a https site that has a valid certificate provides end-to-end encryption, to protect against eavesdropping anywhere along the route. An eavesdropper could still figure out who is talking to whom, but would not be able to decode the conversation without cracking the encryption. Note that https protects against eavesdropping – but does not guarantee that a site is safe. If you visit a hypothetical CriminalsRUs site using a https link, the security would protect you against eavesdropping by other criminals, but not against delivering yourself into the hands of the ones running CriminalsRUs.

Any banking app worth its salt should be using end-to-end encryption on whatever network connections it makes back to your bank's servers. Otherwise the banking app and your bank's servers might as well be writing out your sensitive information on the back of postcards.