Given 0 is a number and division is an arithmetic operation, 0/0 is an arithmetic operation performed on two numbers, which yields a non-number result.
If two numbers in arithmetic operation can yield non-number results. Which other non-number results can come from such operations? Any way of identifying appropriate type of result?
What arithmetic operations can yield a result of a different type than the types of operands?
I think you might have some misunderstandings.
Division is a function $\div\colon \Bbb{R} \times \Bbb{R}\setminus\{0\}\to \Bbb{R}$. This means that it takes in one number from $\Bbb{R}$, and another number from $\Bbb{R}$ that's not $0$ and gives you a number from $\Bbb{R}$.
$0\div 0$ is not defined. That does not mean that it is some mystical "other" thing with the property called "undefined". It just means the function "$\div$" is only defined on some subset of $\Bbb{R}$.
$0\div 0$ does NOT yield a "non number result", it is just a meaningless string of symbols.
If you want to rephrase the question in a more rigorous way, are there more functions on proper subsets of $\Bbb{R}$?
Of course, and they become arbitrary/trivial pretty quickly. For example $\log(x)$ is a function $\log \colon \Bbb{R}_{>0} \to \Bbb{R}$.
I can define a new function, called $+'$ where $+'\colon \Bbb{R}\setminus \{2\} \times \Bbb{R}\setminus\{2\}\to \Bbb{R}$ where $x+'y=x+y$. In this case $2+'2$ is NOT defined.