A unique factorisation domain is an integral domain $R$ which has unique factorisation. That is, every $x\in R$ has a unique decomposition into the form $\prod p_i$ where the $p_i$ are prime, up to order and units.
I find it quite strange that there is no analogue of this concept in group theory. Usually, concepts in ring theory use the addition that is defined in the rings somehow. But in this case, the definition of unique factorisation seems to be a purely multiplicative concept, with no reference to addition whatsoever. So why do we not define the concept of unique factorisation in, say, groups as well? A possible reason I can think of is that if we just took a UFD $R$ and threw away addition, then $(R,\times)$ might not be a group. An unfortunate example of this is that the integers are not a group with respect to multiplication, so it would not be a "unique factorisation group", even if we were to define the concept.
But this does not seem convincing enough of a reason to me to completely abandon the idea. Even if we really wanted to include $\mathbb Z$ inside our new concept of a unique factorisation "group", then why not weaken the group axioms to not require inverses? Still, we do not need to introduce a whole new operation of addition to make sense of the concept. What am I missing here? Why isn't unique factorisation defined this way? What does defining addition contribute?
At least in the study of infinite, finitely generated groups, there is something of an analogue! Many such group theorists like to think about their groups as metric spaces, where the metric is the word metric: if you have a finite set $S$ of generators for a group $G$, the word length of an element $g$, $\|g\|_S$ is the minimum number of elements of $S$ and their inverse present in an equality $g = s_1s_2\dots s_\ell$. The word metric is defined as $$d_S(g,h)=\|gh^{-1}\|_S.$$
The analogue of a UFD in this context is a group $G$ and generating set $S$ with respect to which you have a language of geodesic normal forms, that is, for each $g$ you have a nice expression of $g$ as a product of elements in $S$ and their inverses with minimum (geodesic) word length.
An example is $\mathbb Z\oplus\mathbb Z$ with generating set $\{(1,0),(0,1)\}$. There the normal form for $g$ amounts to expressing it in coordinates as $g = (m,n)$ for some integers $m$ and $n$.
Groups with languages of geodesic normal forms (or various notions close to this) are often a little easier to get a handle on algebraically, making them nicer to study. But, I want to point out that this is a much weaker condition than that of a UFD. For example, there's no analogue of the Chinese Remainder Theorem for all groups with this property.