I am a small business user who is very new to Office365, so please provide your answer in as basic terms as you can... (although I do work in IT, just very limited knowledge of this side of things)
My wife is migrating her two businesses to Office365. Currently we have one Office365 Business Premium account which she plans to use to run both businesses, including emails etc.
For Business 2, she wants to create a Teams forum for people to collaborate (all will be Guests, outside of the organisation), but the Office365 account has the organisation name of Business 1, as this is the core business.
This is creating an issue, as when people for Business 2 join the Teams channels, it's referring to Business 1 on the signup pages, when they're in Teams etc
Is there any way to create a second organisation name to show to those who are joining Teams for Business 1 and Business 2 without creating a second Office365 Business Premium account for business 2?
No, you will need another tenant if you want the name to be reflected correctly on all places. You can easily configure separate teams for the different orgs, but the sign-in pages and URLs will still reflect the initial tenant.
Perhaps she can try a Teams free one for the other company?