I had the same problem so I googled and found this in Microsoft's help pages:

You can add multiple custom domains in all Office 365 subscriptions[...]

This help article looks as if it were possible but they only talk about the business versions and Office 365 Home isnt mentioned with a single word - so I continued my research, even contacted Microsoft - they had no idea, but then I finally found this post:

yesterday I called MS Support and the support engineer was discussing in the background with a colleague. Then he said that it's possible to use own domain with O365 Home... I just had to create a new account with my personal email adres/domain... after that I could order O365 and configure it with my own DNS domain name. Then after changing MX records with my hosting provider this would work for me.

anyway... I tried this anyway... I created a new account with my personal domain... then I ordered Office365Home hooked up to my paypal account....but in the end... no options for me to configure this. So I called back.... I got the same engineer... but then he said.. sorry, I don't have enough technical expertise, cause I'm a sales guy.... strange, because I tried to get tech help... The guy gave me a phonenumber to local (Belgium) Microsoft Tech support.... so I called the number. Then I got first or second line tech support on the line, so I expained the situation and the proposed option from his colleague. But I got a negative response.

No, sir, O365 Home does not allow you to use your own domain... why should it, there are not so many families using their own DNS domain, sir...

DUHH !!! What ? I don't understand their response... I times like this... people make their own family website... use a personal domain.... so I believe this is really a stupid reaction from a Microsoft Tech guy. Probably he has his own domain at home ;-) The only option is to choose for O365 Small Business and pay a lot more, because you need to pay for every user... every family member... so that would cost me at least 200€-250€ yearly.... please get serious ! so I give up.... Also Google Apps only has a business solution... also around 200€ yearly. Office365 Home... useless -> go for Office365 SMB... 2expensive ! I'll try again when Microsoft grows up and gets serious.

After all I don't think it is possible, I also found some forum posts and comments in blogs and they all say it isn't possible.

Hi. I'm Brian and I'll try to help.

If you purchase a Personal or Family subscription, the custom domain should still apply. However, as of November 20, 2023, Microsoft will no longer support custom domains with Microsoft 365 Personal or Family. It will be supported with the Microsoft 365 business subscriptions. Custom domains should still work if you have one but you won't be able to create a new one. My personal inclination is to not use a feature for which there no longer is support because if it were ever to break, it would never be fixed and there would be no recourse. It's not that much more expensive to get a business subscription if a custom domain is important to you when using Microsoft 365 and you don't need to be an actual business to subscribe.

I use a custom domain with the free version of Outlook.com. Microsoft (or GoDaddy) is not the registrar. I have my custom domain point to a mailbox that I then forward to my @outlook.com address and I have the custom address configured as an alias for that Outlook.com mailbox. I also have it set as the default From address. If I use the web interface, the From address is my custom address and people receiving my messages see that address whether it's a new message or a reply to an existing message. It doesn't work as straight-forward with a mail client (you would see "on behalf of" in the From address). However, there's a work-around for that as well if you create a send-only account in the client that sends messages you want to show as From the custom address through the real custom domain mailbox rather than relaying through Outlook.com. It can take just a bit more thought about the From address when sending, but I'm so used to it now I rarely make a mistake. I most often use a Chromebook for my normal computer work and since that uses the web interface, the From is always good.

As long as the code you purchase is a genuine Microsoft 365 code and not, say, a code for Microsoft Office Professional Plus, which is often a pirated license, it shouldn't make a difference.