Let us consider homomorphisms between partial algebras as defined in
https://planetmath.org/homomorphismbetweenpartialalgebras
There, an isomorphism from $A$ to $B$ is a
bijective homomorphism from $A$ to $B$
such that its inverse is a homomorphism, too.
I can see that for partial algebras the second condition is necessary. Is the second condition also necessary if $A$ is a total algebra, or can this condition be dropped? I.e., is condition 2 implied by condition 1 when $A$ is total?
Yes, the second condition is implied by the first when $A$ is total. Indeed, if $f$ is a bijective (or even just surjective) homomorphism $A\to B$ and $A$ is total, then $B$ is also total, since for all function symbol $\varphi$ of arity $n$ and all $b_1,\cdots,b_n\in B$ we can find $a_1,\cdots,a_n\in A$ such that $f(a_i)=b_i$, and then $$\varphi(b_1,\cdots, b_n)=\varphi(f(a_1),\cdots,f(a_n))$$ is defined and equal to $f(\varphi(a_1,\cdots,a_n))$.
In particular $f$ is a bijective homomorphism between algebras, and thus its inverse is also an homomorphism by the usual argument.