I found quite a few sources telling me, the linking number of knots can be generalized to zero homologous knots in oriented 3-Manifolds in terms of the cup product. (For example the Wikipedia article on the cup product) My understanding is as follows:
If $x$ is a zero homologous knot in $M$, and $\sigma$ is a 2-chain with $\delta \sigma = x$, then $\sigma$ is a 2-cycle in the knot complement $M-x$. We can then calculate the cup product of the Poincaré dual of $\sigma$ and an arbitrary second knot $y$. This would be in fact an integer and can be called the linking number.
Bredon has shown in Topology and Geometry that the cup product has a geometric interpretation and therefore aligns with the understanding of the linking number in terms of Seifert surfaces in the standard 3-space.
What i cannot find are sources explicitly showing that such a definition would be well-defined, i.e. why would it be independent from the choice of $\sigma$, or why would it be symmetric for two zero homologous knots?