I used to have to log in every visit (iMac s/safari). Without doing anything (that I remember, at least) I now stay logged in. So it is possible, somehow, to stay logged in when using a Mac. Alas, I can't tell you how.
Well since this thread got revived.
For security reasons, I prefer sites that will automatically time out log ins if no activity.
Almost all password managers will log you in with just one click, no need to type in user name and password. So many other benefits, highly recommend using a password manager. Unique passwords for every site.
I use LastPass, good for both macOS and iOS. Free version or premium.