I often feel the need to refer to both systems "Unix and Linux", with the single word. I am not sure if such word exist?
You'll typically see Unix & Linux referred to as described in this U&L Q&A titled: Is Linux a Unix?.
Often times you'll see them combined and referred to as *nix but even this isn't that popular. Linux & Unix aren't the same thing, and so typically they're referred to independently, hence why this site is called Unix & Linux.
This probably covers the topic the best on Wikipedia: Unix-like.excerpt
The Open Group owns the UNIX trademark and administers the Single UNIX Specification, with the "UNIX" name being used as a certification mark. They do not approve of the construction "Unix-like", and consider it a misuse of their trademark. Their guidelines require "UNIX" to be presented in uppercase or otherwise distinguished from the surrounding text, strongly encourage using it as a branding adjective for a generic word such as "system", and discourage its use in hyphenated phrases.1
Other parties frequently treat "Unix" as a genericized trademark. Some add a wildcard character to the name to make an abbreviation like "Un*x" or "*nix", since Unix-like systems often have Unix-like names such as AIX, A/UX, HP-UX, IRIX, Linux, Minix, Ultrix, Xenix, Xinu, and XNU. These patterns do not literally match many system names, but are still generally recognized to refer to any UNIX descendant or work-alike system, even those with completely dissimilar names such as Darwin/macOS, illumos/Solaris or FreeBSD.
No because they aren't the same. UNIX is a trademarked name that is associated with operating systems that pass qualifications in order to obtain that name. Linux is a work-alike system but not the same thing and was derived from Minix which also isn't UNIX.
Similarly, perhaps, one cannot call "soft serve dairy deserts" ice cream just because they are smooth frozen deserts.