How many unital magmas (magma with an identity element) with three elements are there (up to isomorphism)?
My approach:
List out all of the possible 2x2 multiplication tables for the two non-identity elements. There are $3^{4} = 81$ of these.
Extend these 81 tables by adding in the rows and columns for the identity element.
Manually identify isomorphisms, by re-labelling the two non-identity elements.
I found that of the 81 tables, there are 36 isomorphic pairs of tables, plus 9 more tables where re-labelling simply gives the same table again.
Thus, I conclude that, up to isomorphism, there are $36 + 9 = 45$ unital magmas with three elements.
Is this correct? And is there a more efficient approach to obtaining the solution?
This is correct. I would phrase this in terms of group actions, although it sounds like my argument is pretty much the same as yours. Let the underlying set be $M=\{1,x,y\}$. Every unital magma with $3$ elements is isomorphic to a unital magma with underlying set $M$ and $1$ being the unit by appropriately labeling the elements. The number of unital magma structures on $M$ is the number of maps $\{x,y\}\times\{x,y\}\rightarrow M$ (cause the rest of the multiplication is determined by the unit axiom and there are no further restrictions), of which there are $3^{2\cdot2}=81$ many. To determine how many of these are isomorphic, note that an isomorphism of unital magmas necessarily preserves the unit, so the only possible isomorphism is always given by the involution $\tau\colon M\rightarrow M$ that fixes $1$ and switches $x$ and $y$ (and every multiplication determines one that is isomorphic to it via $\tau$ by transport of structure, i.e. $m\mapsto\tau\circ m\circ(\tau\times\tau)$).
Thus, we obtain that $\mathbb{Z}/2\mathbb{Z}$ acts on the set of unital magma structures on $M$ and the isomorphism classes of such structures are precisely the orbits of this action. The condition that a unital magma structure is a fixed point of this action is equivalent to $\tau(x^2)=y^2$ and $\tau(xy)=yx$, so two of these products can be arbitrarily specified and the other two are then uniquely determined, meaning there are exactly $3^2=9$ such unital magma structures. This yields the overall number of orbits as $9+\frac{81-9}{2}=45$.