When you transfer content to a new phone from another phone it transfers everything, including app data (Apps are redownloaded from the app store). But the VPN settings for the previous phone are customized for that phone, and generally won’t be compatible with the new phone.
But the real fact is that you don’t need VPN for security on an iPhone; all it does is make you less secure (because your VPN provider knows everywhere you go, and can you really trust them, not to mention that they can be hacked), slows down your communications to about 1/4 of the maximum bandwidth, and can introduce communications problems (as you discovered).
Now that Apple has Private Relay there is absolutely no need for VPN; Private Relay is sort of a super VPN that uses 2 links rather than one, so it is closer to TOR in its privacy features. It costs 99¢ a month and is included with 50G iCloud+ (and up), which is less than you are paying for VPN. And is compatible across devices and upgrades.