Do wallets save a backup of my private keys and seed?
It depends on what wallet you are using.

Most people prefer to use wallets that do not have knowledge of your private key outside of your machine, such as hardware wallets, electrum, Bitcoin Core. This is for safety, since any provider who has your keys could easily decide to move all your coins to an address that only they control, and rob you blind.

That said, there are some services that offer custodial services, usually in conjunction with exchange services. In fact, every exchange is a custodian keyhold for the deposit addresses they assign you on their platform.

A few years ago, perhaps there was some merit in trusting a reputable custodial service who had the infrastructure to always backup keys. In 2018, BIP32/39/44 can make a backup as simple as writing down a handful of words and keeping it very safely, so maintaining control of your keys is the preferred option.