How to switch between 2 Steam accounts on one PC without having to enter mobile code every time?

You say you both use the same Windows account.

Create a new account for your brother. Login to steam with his account information. Sign in back as your shared account. Create a new shortcut to steam and change the command to

%windir%\system32\runas.exe /user:YourBrothersWindowsAccount /savecred Steam.exe

(make sure to use the full path to Steam.exe from the original shortcut)

Now, when your brother wants to use his Steam account, all he has to do is close Steam, and run it again through this shortcut.

Do note that Steam will use the new windows profile, so if your brother has games that store configuration or saved games in the user profile, he will need to copy them over to the new account.

The @Luaan's answer seems like a cool workaround. It suggests to create a "fake" second Windows account and don't use it. I think that you should take one step further, and actually use that second account! Don't you have similar issues with other stuff, like web browser and Facebook, or Skype? Don't you need to endlessly log-in-and-out from various apps and services? The popular way for using same PC by two different people is to have separate Windows user accounts. It's really quite a lightweight solution and I think that nowadays you will be even able to switch profiles without closing your open windows. You can share any data you like, it's not an issue. You will have your own wallpaper, your own shortcuts, you will be always logged in to your Facebook, Skype, and the most important, Steam.

Summing up: to switch between 2 Steam accounts on one PC without having to enter mobile code every time, just switch the Windows account.

I got it working thanks to Reddit u/psxsquall.

The way around it works by using Steam's Big Picture Mode.

You use it and login with both accounts and tick the "remember me" button in Big Picture Mode, confirm with the phone codes once, and then when you log out (only through Big Picture Mode) it allows you to switch users without asking for password/phone codes.

TL;DR: Use Big Picture Mode to sign out, but make sure never to sign out from the normal Steam interface (you can still exit steam normally) and only through the Big Picture interface.