No, an eruv affects only an area’s status for the purposes of carrying on the Sabbath and Yom Kippur. This is no different that how every inch of the continental US has been designated part of a specific Catholic diocese, governed by a Bishop for the purposes of Catholic religious observance. Much like the eruv’s “private domain”, being placed in an diocese does not affect you unless you are practicing that particular religion.
The use of the term “private domain” when talking about eruvin might cause one think that its construction implies some sort of rights over the properties inside. There is no basis for this view in US law or Jewish law.
The class of “domain” here refers to how enclosed an area is, not who has property rights over it. The only function of an eruv is to make an area technically “enclosed” and allow carrying inside it in public places as if it was not the Sabbath or Yom Kippur. An eruv does not give anyone permission to occupy, sell or trespass on another’s private property.
There are also those that object to encountering any sort of religious “domain”, but we think it’s apparent that such a designation is entirely imaginary to anyone but Orthodox Jews.