Processes in Microsoft 365 for setting up Office apps, redeeming product keys, and activating licenses.
Nothing odd, one's a business account the other a home account. You can't mix them.
What you should do is have her install upgrade to Office 365 Home, assign your home account (not business account) an Office 365 Home license (using one of the 5 she can share). That way you and she can use Word, Excel, Etc.
Then in Outlook just add your business email account to access it.
It's how my setup at home works with home users and my work account.
Yes, the root problem is that Office 365 "Business" and "Consumer" (like 365 Personal) do not "play well together".
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365 Personal and 365 Home are the same thing except that 365 Home allows more people to share it. So I don't see how "upgrading" to Home from Personal is going to help with this problem.
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The root of the problem is that Windows is trying to let her use the Office Business installed programs with her Personal license. MS is really lazy when it comes to Office installations. They put everything into the same place (which causes problems if you tried to mix 32 and 64 bit installations). But it is also a problem if Windows gets confused to tries to have your wife run the "wrong" Office installation.
Did you install Office Business using the Windows Admin account?
You could try having your wife re-install Office Personal inside her personal Windows account. If you are lucky, she will have access to her 365 Personal installation and you will still have access to 365 Business. If you are not so lucky, you'll start having problems.
Another option would be to have your wife run a "virtual machine", a "VM", like Windows Hyper-V which is part of Windows Pro (or Enterprise), or the free "VirtualBox". The VM creates a "virtual computer" inside of Windows that is isolated from Windows. That would allow you to run the 2 conflicting versions of Office. It is a big gun to deploy, but welcome to "Windoze" and MS world.