I'm running 10.13.6 on a late-2014 MBP. After the latest Safari update (to 12.1), bank websites don't recognize my computer and/or autofilled user name+password. Facebook says it doesn't recognize my browser, so it asked me to implement the second-half of two-factor authentication; I did this, and can now login to that site.
I don't have this issue logging into lower-security websites, like discussion forums, nor with sites for which I have a persistent login (i.e., those I can use without needing to re-login).
Is this standard behavior following a browser update, such that sites suspect I'm using a different computer or browser? I don't recall this happening with previous Safari updates (but maybe it did and I simply don't remember).