The main difference between amalgamating groups and semigroups is that
amalgamation is not always possible for semigroups.
One of the simplest examples (from Lyapin's book) is the following.

Take two commutative semigroups (which are even groups):
$$
A=\{\dots,{1\over16},{1\over8},{1\over4},{1\over2},1,2,4,\dots\}
\quad\hbox{and}\quad
B=\{\dots,{1\over16},-{1\over8},{1\over4},-{1\over2},1,2,4,\dots\},
$$
where the operation in $A$ is the usual multiplication and the operation in
$B$ is $x\circ y=\pm xy$, where the sign is chosen so that $x\circ y\in B$.

It is easy to see that the amalgam (where
$A\cap B=\{\dots,{1\over16},{1\over4},1,2,4,\dots\}$)
cannot be embedded in a common semigroup. Indeed, in such a common semigroup, we would have
$$
{1\over2}={1\over2}\cdot1={1\over2}\cdot(2\circ(-{1\over2}))=
({1\over2}\cdot2)\circ(-{1\over2})=1\circ(-{1\over2})=-{1\over2}.
$$