In addition to the fantastic answers by @leocwlau and @yuhuisg; in most cases (Safari may not function this way), but in general, when I have tested multiple different domains in our infrastructure, without using any fancy cross domain parameters (in fact, I will open the sites in different tabs - not linking from within siteA to siteB), due to the Demdex cookie, I am set with the same ECID on all domains...
Again, this has a potential to fail due to 3-party cookie usage and the cookie-pocalypse...
If you cannot pass query string params, I wonder if you have any way to do some sort of server-side custom solution. I hear what the others are saying about using Adobe's solution instead of your own.. but if it's not an option, its not an option... so you may have to think outside the box... it will be better than nothing,
One solution we did a few years agi (this was within the same site), but we created replication ECID server side cookies.. if the ECID cookie was lost/deleted, we would re-write it from the server side cookie. I wonder if there is something you can do to identify the user on your servers and use server side cookies to set first party cookies when the user comes to a site... or send the ECID on click as a post parameter that you can then set the cookies from upon entering the site.
