It depends on the circumstance but no - the client does not outright own all of the work done on their domain simply for having paid their invoice. The client owns the site itself and much of what we provide on the site but in many cases the programming that goes into the backend or design elements has been developed within our company, customized for the client, and is licensed to the client for their use. Many of the 'structures' we utilize are based on systems of code that are unique to our company and are ultimately very valuable to us (if for no other reason than the amount of work that went into them!

).
If it was owned by the client then they'd have the 'right' to populate other domains with the same design, same program, same code etc. and could potentially profit from proliferation of our developments without compensation to us - this is obviously not to our advantage as developers. When purchasing Adobe or Microsoft software, you pay for a license for ONE computer - multiple use requires additional licensing. (We do offer additional licensing for clients who need to utilize a design or program on more than one URL.)
Another example of how this works is to point to the software/code that runs this forum - I purchased a license to run it on my (single) domain - this allows me to customize the forum, use it as I like, but not to copy or utlize the code in part of in full on any other domain (or subdomain). If I want to create another forum I need to purchase another license.
I think the confusion may come with the terminology used - we are developing, in many cases, software - not just a website...Does that help clarify?
Now, if a client paid us to develop a software FOR them that would be a different story, and would carry it's own specific type of agreement (and fee scale)