With keys such as a U2F key, you can use it for multiple sites (each site will have a site-specific keypair). Most sites allow you to associate multiple U2F keys as well. Since U2F keys typically cost 7 to 15 Euro, it's really not much of a problem to buy one for use and to put one as a backup in a fire-proof safe.
That said, I think it's bad to rely on one factor. If the key gets stolen, security is compromised. The combination of a password and a security token is far more secure. People should just stop memorizing passwords for every site and e.g. use a password manager.
A lot of banks already give them away to their customers because the hardware costs are less than the fraud costs associated with compromised accounts. And in an ideal world, said keys would be general use, not just for logging into the bank's site.
That said, I think it's bad to rely on one factor. If the key gets stolen, security is compromised. The combination of a password and a security token is far more secure. People should just stop memorizing passwords for every site and e.g. use a password manager.