Running 2 instances of Steam with different accounts on the same PC

You need to create a new local windows user to run the second Steam instance.
Here is how to do it in Windows 11, 10-chads are on their own.

In Windows, go to

Settings > Accounts > Other users

Under Add other user click

Add account

Below the E-Mail input field, click

I don't have this person's sign-in information

Again, below the input field, click

Add a user without a Microsoft account

Enter a username, I've used BA but you can be yourself. Enter a password and fill out the stupid security questions.

Now create a bluearchive.bat file somewhere with the following contents (without the line numbers):

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set VPROJECT=steamBA
runas /user:BA "\"C:\Program Files (x86)\Steam\steam.exe\" -master_ipc_name_override steamBA -userchooser"

Replace the BA in /user:BA with the name of your other windows account. For example: if you called it Sensei you would put /user:Sensei

Replace C:\Program Files (x86)\Steam\steam.exe with the path to your Steam installation if you changed it. Make sure to leave the "\" at the start and \" at the end, it's for path escaping, don't worry about it.
For example, my Steam is installed under C:\Steam so my command looks like this:

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runas /user:BA "\"C:\Steam\steam.exe\" -master_ipc_name_override steamBA -userchooser"

Now if you doubleclick this .bat file, it should prompt you for the other users windows account password, type it in and hit enter.
The command window will close and a new Steam window should pop up with an account selector.

Login to your second Steam account and play the game.