I'd like to understand if there could be disadvantages to submitting applications to several graduate programs within a single university system (e.g. Texas A&M, University of California, etc)? May there be any cross-communication between different campuses/universities within a system during the admission process? Are the admission decisions always made independently? If my application is rejected by one university within a system, is it more likely to be rejected by other universities within the same system? The subject is engineering, but I am not sure how much this matters.
"Always made independently" is too strong. There might be some cross communication, but I doubt that it is common or determinative. You are evaluated for admission by some committee from a specific department at a specific campus.
There would be little reason for a faculty member at one campus to need/want to comment on an application to another, as they wouldn't be dealing with you in any case.
And, for most systems you can't automatically transfer between campuses anyway, so the decision is local. It is even likely that the "standards" differ slightly at the different campuses in a large state system. And some of them are in competition with one another, and not just in football.
So, there are unlikely to be any side effects at all.
To the extent of my experience, in the U.S. it would be a violation of privacy laws if there were any of the sort of inter-campus communication you suggest. Even in the "bad old days" when there was waaaaay too much under-the-table chatter, it was not between campuses of "the same" university. I know of not even one instance of this in Minnesota, in math, since 1982 (and, to be clear, I've been involved in grad math admissions, and/or been the Dir of Grad Studies in math, for all those years...)
If anything, I'd suspect that various campuses of the same university systems might be more independent of each other, since they do compete to some degree.