The informal analogue, is simply the notion of *topos of sheaves.*

If I work in a "ground" topos (whose object I call set), then a "forcing extention" would be just a Grothendieck topos, that is a topos of sheaves on a small site.

If you want to adopt an external point of view and start from an elementary topos $\mathcal{E}$ , then **a forcing extension of $\mathcal{E}$ is a topos $\mathcal{F}$ that can be obtained as the category of $\mathcal{E}$-valued sheaves on an internal site in $\mathcal{F}$**, where internal site means " a category object in $\mathcal{E}$ endowed with a "topology".

The simplest way to define the word "topology" here is to say that it is a Joyal-Tierney operator in the topos of $\mathcal{E}$-valued presheaves on the category object. But one can also define it in a more Grothendieckian style using collection of subobjects of power objects satisfying the internal version of the axioms of a topology.

It is a well known theorem of topos theory that the topos that can be obtained from $\mathcal{E}$ this way **are exactly the topos endowed with a  bounded geometric morphism $\mathcal{F} \to \mathcal{E}$.** (see section B3.3 of P.T.Johnstone Sketches of an elephant).


The best way to get a feeling of why this is a good analogy is to look at the topos theoretic proof of the independence of the continuum hypothesis in MacLane and Moerdijk "Sheaves in geometry and logic" (section VI.2).


**However, it is not a perfect analogy:** while forcing in set theory people consider a model that contains a "generic filter", in topos theory we consider a model that contains "the universal filter" (in the sense of classifying toposes). The point here is that the toposes obtained this way are not well-pointed in general, so they can not directly corresponds to a model of ZFC.

If you want a more precise analogy you need to combine the construction of the topos of sheaves with a construction that reproduce a model of ZFC out of a topos. For this I recommend to look at Mike Shulman's [paper][1] that give a very good exposition to the topic. 


  [1]: https://arxiv.org/abs/1808.05204v2