There is a natural functor from Ring to the category of groups, $\mathrm{Grp}$, which sends each ring $R$ to its group of units $U(R)$ and each ring homomorphism to the restriction to $U(R)$.
I have never heard of a "natural functor" before and haven't found it online. Is this simply a natural transformation in the category of categories (where functors are morphisms)?
If so, how do we formalize the notion that the above mentioned functor is natural?
To be clear, the use of the word "natural" here is informal, not formal. Loosely speaking it means that the definition of the group-of-units functor is a "natural thing to do" to a ring, in that it only uses the ring axioms in a very basic way (namely that multiplication is associative and unital).
It's unfortunate that we also have a formal notion of naturality in category theory which makes it hard to distinguish between the formal and informal uses of the word.
However, in this particular case it is actually possible to upgrade this informal naturality to something formal, as follows. We can actually define rings and groups internal to other categories, more specifically in any category with finite products, by rewriting all of the ring and group axioms in terms of the commutativity of certain diagrams. For example a topological group is a group internal to $\text{Top}$. If $C$ is a category with finite products write $\text{Ring}_C$ for the category of ring objects in $C$ and $\text{Grp}_C$ for the category of group objects in $C$. Then, if we additionally assume that $C$ has equalizers, there is a group-of-units functor
$$U_C: \text{Ring}_C \to \text{Grp}_C$$
sending a ring object $R \in \text{Ring}_C$ to the equalizer of the three maps $R \times R \to R$ given by the multiplication $m$, the reversed multiplication (in the other order), and the constant map $1$ equal to the multiplicative identity on the other (this cuts out the locus $\{ (r, s) \in R^2 : rs = sr = 1 \}$), equipped with a suitable group object structure. This collection of functors is moreover natural in $C$ in the technical, formal sense (in that a suitable square commutes involving $U_C$ and $U_D$ if we have a functor $F : C \to D$ which preserves finite limits), although since $C$ itself is a variable that ranges over categories this all has to be understood in a suitable 2-categorical sense.
This is one way of formalizing the informal idea that we are "only using the ring axioms in a very basic way" when we define the group of units; loosely speaking the existence of the natural (in the technical sense) collection of functors $U_C$ above says that the definition of the group of units only involves the "essential concept of a ring" (that is, the notion of a ring object) and is not sensitive to details like what category we happen to be considering ring objects in.
(Some care needs to be taken here to interpret $U_C$ correctly. Its existence does not imply, for example, that the group of units $R^{\times}$ of a topological ring $R$ is a topological group when equipped with the subspace topology; in general inversion can fail to be continuous. Instead we are topologizing $R^{\times}$ as the $\{ rs = sr = 1 \}$ subspace of $R \times R$. Here inversion is just given by switching the coordinates so it is clearly continuous, which means this topology differs from the subspace topology $R^{\times} \subset R$ in general.)
In this context the word "natural" does not have a mathematical meaning. That is, there is no mathematical property of functors called "naturality".
But why does the author use the word "natural"? He wants to say that there is an obvious or natural way to define the functor in question. This is of course informal language use which suggests that "everybody should see the obvious without needing explicit and lengthy explanation".
In that sense you find the word "natural" quite often in the mathematical literature, but you will not find a mathematical definition.
There are a few more words of the same spirit, the most prominent example being "canonical". When you see it, you should be aware that it usually does not have a mathematical meaning, but also expresses the attitude "this is so obvious, it's not worth to spend time to give further explanations"
A "Natural" functor is typically a functor that arises from a natural mathematical situation, ie. $U:\textbf{Ring} \rightarrow \textbf{Grp}$ described as earlier, $F:\textbf{Ring}\rightarrow \textbf{Ab}$ sends a ring to its underlying abelian group, and morphisms to abelian group homomorphisms.
Various constructions can be described in terms of functors, which was the motivation for Category Theory to begin with.
I'm writing this as an answer because it is too large to be a comment. In your specific example there might be various notions of "naturality" under consideration.
First of all, you might consider the category of group objects in the monoidal category $(Set,\times)$, which is simply the category of groups $Grp$. There's a "natural" functor from the category of $(Set,\times)$-group objects to $Set$ which simply forgets the group object structure. This notion of naturality works in many other examples, such as monoid objects in a monoidal category, Lie algebra objects in a symmetric monoidal category, etc. Repeating this process in $Grp$ we obtain the category of $(Grp,\times)$-group objects, which is simply $Ab$, the category of abelian groups. Again there is a "natural" (in the sense mentioned before) functor $Ab \to Grp$. That yields us a "natural" composition $Ab \to Set$. Read more about group objects here.
Now, we can consider the categories of monoid objects in the monoidal categories $(Ab,\otimes)$ and $(Set,\times)$. That gives us respectively the categories of rings and ordinary monoids. The functor $Ab \to Set$ mentioned previously induces "naturally" a functor $Ring \to Mon$. Read more about that here.
Finally, there's the functor $Mon \to Grp$ which sends each monoid to its group of "units" and each monoid homomorphism to its restriction, but I'm unable to find any references on this kind of construction.
