I had that last week week when I travelled with my laptop, also after getting home I think I had to type the password once. Now it isnt asking for it anymore. I did not see it on my stationary.
Do you connect from different networks? Maybe its a new security measure.

EDIT: I misunderstood, I thought this was about logging into the Steam Client, that is what I did on my laptop

Same network, same IP, same computer, same browser, same everything.

I go to the site, click login. Tick "stay logged in" and "remember username", then click the "sign in with steam" button. I'm redirected to steamcommunity.com for confirmation, then I'm logged in.

I close the browser, reopen it, visit the site again, and I'm logged out. That's not how it worked previously.

Yeah sorry about that, I have confirmed it's a bug with some code changes. Re-working a bunch of stuff atm, will be fixed in a day or two hopefully. Thanks for finding the issue.

Oddly I do not have this issue in spite of regularly logging in from 3 different locations

This should now be fixed, all social logins keep you logged in until you log out.