That's the purpose of a two-factor (i.e. mobile) authentication.

The workarounds I am aware of is to either:

• Remove the mobile authentication entirely, or
• Use the 'Family Share' function to use games from both accounts, while only logged in to one account.

It seems there are some better workarounds other than these two but these are also viable alternatives.

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.

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.

Luaan's answer is great, but things have changed a bit.

On Windows 10

1. Create a new user in Windows
   • Settings > Accounts > Family & other users > Add someone else to this PC > I don't have this person's sign-in information > Add a user without a Microsoft account
   • Choose a one-word username for simplicity (say, "Bob")
   • Leave the password blank (yes, Windows will let you do this)
2. Login as that user, open Steam, login with the new account
3. Switch back to your normal user. Copy and paste the desktop Steam shortcut so now you have two copies of it.
4. Rename the new shortcut to the new username ("Bob" in this example) so you know which is which.
5. Right click on the new shortcut and select Properties.
6. Change Target to %windir%system32runas.exe /user:Bob /savecred "C:Program Files (x86)SteamSteam.exe"
   • Change "Bob" to your new username
   • If you didn't install Steam in the default location, then change C:Program Files (x86)SteamSteam.exe to wherever Steam is installed on your computer. If you're not sure, look at the properties of the original Steam shortcut.
7. Tell windows to allow blank passwords:
   • Search for regedit in the Start Menu. It's also called "Windows Registry" or "Registry Explorer"
   • Copy and paste this into the address bar at the top: ComputerHKEY_LOCAL_MACHINESYSTEMControlSet001ControlLsa
   • Double click on LimitBlankPasswordUse on the right
   • Change the 1 to a 0
   • Close the registry editor

Now when you double click on the new Steam shortcut it should be logged in with the second Steam account, and the original shortcut will still be on the first account.

You can't have both open at once, and if you need to switch between the two you have to completely exit Steam by clicking Steam > Exit, or by right clicking the Steam icon in the Windows taskbar (bottom right) and clicking Exit.

You can use PsTools (by Microsoft) to achieve this in a similar method to those already given. However, PsTools allows you to embed another users Windows password, so you don't have to hack the registry (which will likely be reset to it's default value in future Windows updates, breaking the shortcut).

0. Create a new user, with a password
1. Head to: https://docs.microsoft.com/en-us/sysinternals/downloads/psexec
2. Download the zip file, and unzip it somewhere awesome, say: C:Program FilesPsTools
3. Hold WinKey + Push R
4. Type: SystemPropertiesAdvanced, and click OK
5. Click Environment Variables
6. Under System variables, scroll down until you find Path, under the Variable column, click it, and then click Edit... below
7. Click New, and type in the awesome location that you unzipped PsTools to (C:Program FilesPsTools - In this case), then click OK in each of the windows that opened
8. Now... Right-click on the desktop, and select New ~> Create Shortcut
9. In the Type the location of the item: textbox, type:

psexec -user NewUserName -p NewPassword "C:Program Files (x86)SteamSteam.exe"

Replacing NewUserName and NewPassword with the credentials you used in part 0, and changing the steam executable path to your steam installation (the example above uses the default installation location)

10. Click Next
11. Type a meaningful shortcut name in to the box, and click Finish
12. Double click the shiny new shortcut
13. Enjoy!

There's one caveat to this, and that is that it opens a CMD shell window, this can be hidden by using some VB/C#/PowerShell stuff, but it would significantly complicate this one minute set of instructions. The window can be closed immediately, and steam will continue running just fine.

There's an even easier way. Using a Steam account switcher. Most let you swap accounts easily, some without even needing to enter a password or 2-factor again. Most of these are open-source, including SAM - Steam Account Manager (180★), Steam Account Switcher (75★) and Steam Account Switcher (42★).

I am biased here, as I have created a solution myself. Not just for Steam, but also Origin, Epic Games and a ton of other platforms, hence the other suggestions first. It swaps files on your computer that store the 'last logged in account', or token, and allow you to swap without entering a password or 2-factor again (Assuming you've logged in at least once before - and don't click "Sign out"). See the TcNo Account Switcher (201★)