Outlook 2016 doesn't accept password or app password

It doesn't help much. If you look at your access history, you will likely see a mixture of IPv4 and IPv6 accessing your account, even on the same computer at the same time. The server doesn't really know if these are the same device or different devices.

Interesting sort of workaround... if you open outlook.com in your browser and log in, Outlook 2013 will stay connected and let you use it for your hotmail account...

Funny fact, I travelled overseas back in 1/23/2024 when this issue started and I am now in a 2nd same international trip and the issue is still not fixed in literally 2 months! What a sad state!

As far as the error is concerned, for desktop Outlook app, you can just copy/paste the password into the popup and enter. Do that a few times and cancel the popup and your data will sync. I think Outlook authenticates and challenges for each data session and there are multiple sessions when you use Outlook normally.

For mobile EAS connections, i.e., Gmail accessing Hotmail via EAS, you are mostly of out luck. The popup prompt will generate a notification and tone, which is very disruptive. There is nothing you can do but to ignore it. Every now and then when sun moon star and Microsoft's evil all line up, your data will sync but it's sporadic.

Now onto "new Outlook" clients. I setup some sealed off dev environments to monitor data traffic, and I can confirm the "new Outlook", Outlook for Android, Outlook for iOS.... will take your non-Microsoft data. When you add those accounts into Outlook, you are not just storing your credential in the app, you are giving it to Microsoft and it uses it to access all the data.

In a nutshell, if you have non-Microsoft data that you want to protect, DO NOT USE the new family of Outlook apps. You can use it for Hotmail or other Microsoft services. Since MS already have those data, I assume minimal additional harm. However, if you want your Outlook mobile app's data accessible in another phone apps, such as the Phone's dialer, then you have enable the sync to that data, and it's a 2-way sync. Microsoft will then suck your data on other platforms that you have added to your phone. Basically, Outlook mobile apps acts as a proxy to all of your other data, even ones you didn't directly give to Outlook. This is one of most invasive apps I have seen in a long long time.

I'm the same - mail getting through but with the pop-ups.

I'm running 2013 as well.

While I'm certainly experiencing issues, I have discovered some oddities that may help you manage your situation.

On the "Send/Receive" tab, if you have the "Type Exchange Password & Connect" click that.
Enter the password.
It should accept it through that Feature (it usually does for me).
Click "Update Folder"

Afterwards, if you get the dialog box fired, just cancel or X it to close it.
This has happened to be thousands of times but even though I cancel or close it, mail is still working.
If you see the message on the status bar, many times I can just click it and it get the stored password and that usually works for me.