Let $G$ be a magma, $\sim$ a congruence relation, $G/\!\!\sim$ the corresponding quotient magma and $\pi_\sim: G\to G/\!\!\sim$, $x\mapsto [x]_\sim$ the canonical epimorphism. Assume that $F$ is another magma with a homomorphism $f_\sim:F\to G/\!\!\sim$.
Can I somehow deduce that there generally exists a homomorphism $f:F\to G$ with $\pi_\sim\circ f= f_\sim$?
If not, what would be a counter example?
Are there more general for statements for (relation, algebraic,...) quotient structures like this?
No, you cannot deduce that. For a simple example, take $G$ to be the positive natural numbers under addition, and let $\sim$ be the total congruence, so that $G/\sim$ is a one-element semigroup/magma. Now let $F$ be a one element magme. Then there is trivially a homomorphism $F\to G/\!\!\sim$, but there is no homomorphism $F\to G$ that induces the map, because that would require mapping the single element of $F$ to an idempotent in $G$, and $G$ does not have idempotents.
In general, if $F$ has the property that every time you have a map $f\colon F\to G/\!\!\sim$ there exists a map $g\colon F\to G$ such that $f=\pi_{\sim}\circ f$, then $F$ is said to be projective (in analogy to this property for modules).