In topology, we can speak of the "[topology induced by a function][1]", which is the (coarsest) topology on $X$ such that $f:X\to Y$ is continuous, given a topology on Y, or vice versa.
I can imagine that there are similar notions of "induced group structure" which gives a set a group structure such that a given function is a group homomorphism, and an "induced vector space structure" and so forth.
Is there a general (categorical) notion of this? https://en.m.wikipedia.org/wiki/Induced_topology
This is actually dramatically different in algebraic than topological situations. If $A$ is equipped with some algebraic structure and $f:A\to B$ is a function, $B$ may not admit any algebraic structure for which $f$ is a homomorphism. Indeed, if $f$ is surjective then such structure is well known to be unique when it exists: $f$ becomes a quotient map, and the operations on $B$ are induces by the operations on $A$. But it doesn't always exist, as the equivalence relation induces by $f$ must respect the algebraic operations of $A$. For instance if $A$ is a group and $f(a)=f(b)=f(e),$ we must also have $f(ab)=f(e)$, which an arbitrary function will not satisfy.
If $f$ is not surjective then the partially ordered set of algebraic structures on $B$ making $f$ into a homomorphism may have multiple elements, but it's discrete, at least if the partial order is defined using homomorphisms given by the identity function on $B$. This is the key difference with topology: the identity map can be continuous without being an isomorphism, which is impossible in algebra.
There is a general theory here, although as I explained above it doesn't apply to your examples. This is the theory of topological concrete categories. Important examples are not that abundant, but uniform spaces and measure spaces are a couple beyond topological spaces.