Yes, the Adobe Visitor Service should be able to properly identify most users across domains without requiring any special treatment.
For instance, in my network, if I open a new Incognito window and open domain A (and check my ECID), then I open a new tab and open domain B, I will see the same ECID... I can't close my Incognito session since that will fully clear all first and third party cookies... but even if I take a browser that I don't use regularly, and do the same thing (or even close the browser between domains) ... so long as I still have the third party cookies set by the Visitor Id Service (demdex) I should be re-identified as the same user.
Now, that said... both domains still have to be tracking into the same suite for my campaigns to be maintained.. both domains should be treated as internal URLs (so there is no exit tracking or referrer instance being tracked that can cause issues with the flow).
I would first check your implementation that all of the above is properly implemented...
If you are still having issues, you can add in the appendVisitorIDsTo plugin (https://experienceleague.adobe.com/en/docs/id-service/using/id-service-api/methods/appendvisitorid) that should append the ECID from domain A to domain B in the flow (this should compensate for potential loss due to third party cookie rejection)...this will append query parameters to your URL when you move from domain to domain, but you have to make sure that domain B is reading this information and using it... (I've heard of people mentioning that this wasn't always happening for them.... I don't use this myself, since I haven't had a lot of issues thus far)
Again, for your campaigns to properly work, both sites need to track into the same suite.. because even if the same user is identified, the campaign is limited to the suite where it was captured.
Hope this helps, at least as a starting point for investigating your implementation and what is happening.