I click on my account picture in the top right, then click on "Add account" and log into the other one. Presto, two accounts.
Disclaimer: I am about 99% certain this is not some weird internal Google feature (not on the gmail team, don't worry), but it's also been > 1 week since I had to go through this performance so I may have forgotten some detail.
And then when you click on some Google service it always logs you in with a wrong account. Good luck trying to add something to calendar by clicking to a link.
I assume this is hyperbole (I'm not sure what magic you're expecting...maybe something like intents where it asks you every time which account to use?), but the logic isn't complicated. It just uses the first account you logged in with. I'm usually signed in to two accounts (and have a gmail tab for each open right now), so I just sign in to the one I do my primary calendaring, etc in first.
If you really want to keep them separated, you should just use Chrome's multi user accounts or their equivalent in other browsers.
Actually, you explained the problem without noticing it: "It just uses the first account you logged in with." Maybe it should be default account? Because "first account" will always change in round robin fashion because cookies will expire first for the newer account (which is second) and then after you login it will become first.
I've just had a quick test and it actually does okay; clicking on "calendar" opens the calendar for the account I'm currently looking at.
I guess it might not work well if you click a random link somewhere with two accounts open simultaneously, although I have a bit of sympathy for them there because that does seem kinda tricky to solve.
Apart from Google's multiple signin, Chrome supports account profiles, so you can have two Chrome windows with completely different cookies, logins, everything.
1. Sign in to personal Gmail by going to gmail.google.com
2. Sign in to work Gmail by going to mail.google.com/a/foo.com (obviously, replace foo.com with your work domain)