These are divisible semigroups and uniquely divisible semigroups respectively. The reason you cannot find anything about them on the internet is that almost nothing is known. Say, every idempotent semigroup is obviously divisible and uniquely divisible. Every divisible infinite semigroup $S$ is not residually finite unless $S$ consists of idempotents. Indeed, if $H$ is a finite homomorphic image of $S$, then for some $n$ $x^n$ is an idempotent in $H$ for every $x$. Now take any $y\in S$. It has a root $a$ of degree $n$. Hence the image of $y$ in $H$ is the image of $a^n$, hence an idempotent. Thus $y$ cannot be separated from $y^2$ in any finite homomorphic image. Thus if $S$ is residually finite, it must consist of idempotents.

These are divisible semigroups and uniquely divisible semigroups respectively. The reason you cannot find anything about them on the internet is that almost nothing is known. Say, every idempotent semigroup is obviously divisible and uniquely divisible. Every divisible infinite semigroup $S$ is not residually finite unless $S$ consists of idempotents. Indeed, if $H$ is a finite homomorphic image of $S$, then for some $n$ $x^n$ is an idempotent in $H$ for every $x$. Now take any $y\in S$. It has a root $a$ of degree $n$. Hence the image of $y$ in $H$ is the image of $a^n$, hence an idempotent. Thus $y$ cannot be separated from $y^2$ in any finite homomorphic image. Thus if $S$ is residually finite, it must consist of idempotents.

There are infinite finitely generated divisible and uniquely divisible groups. The first example was constructed in Guba, V. S. A finitely generated complete group. Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 5, 883–924.