Let $A$ be a category with one object $\star$. Then as far as I understand, the unique representable functor is $A(\star,-):A\to Set$. Isn't this representable functor genuinly unique, not just up to isomorphism? If not, what is an example of two distinct (but isomorphic) representable functors?
Also, is there an explicit description of left regular representation? How to obtain it?
The set you get when applying $A(\ast,-)$ to $\ast$ is the underlying set of $\mathrm{End}(\ast)$. But there's no requirement that a functor creates that kind of set. For instance, if $B$ is any one-element set in $\mathsf{Set}$, then $B\times A(\ast,-)$ is not technically the same functor as $A(\ast,-)$, although there is a clear natural isomorphism $A(\ast,-)\cong B\times A(\ast,-)$ given by $x\mapsto (b,x)$.
If $M$ is a monoid, the left regular representation means the action of $M$ on itself given by the binary operation on the monoid. That is, the monoid operation $M\times M\to M$ is the action. The image of the functor $A(\ast,-)$ then would bet the set $\mathrm{End}(\ast)$ and each morphism $f:\ast\to\ast$ would turn into a function $\mathrm{End}(\ast)\to\mathrm{End}(\ast)$ given by $g\mapsto f\circ g$.