EDIT:
Regarding the original question: I think that ultimately the sense in which functor calculus is like stacks is that we are peforming localizations of a functor category. Moreover, these localizations are left exact. This leads to similarities in how the localizations are described (via descent with respect to certain "covering" diagrams, which can be described as (hyper)covers if you like) and how the localizations are constructed. More on this below.
Regarding the analogy between manifold calculus and Goodwillie calculus: In both cases, we have a tower of localizations of a functor category $Fun(C,D)$. In both cases, we have natural descriptions of the localizations in terms of descent with respect to certain "covering" diagrams, and in both cases the tower of localizations corresponds to a filtration of the collection of "covering" diagrams for the strongest localization.
But I don't think these filtrations arise in analogous ways. In the manifold calculus, the strongest localization comes from the usual Grothendieck topology on the site of manifolds: "covers" are Cech covers in the usual sense; the $n$th localization is at the subcollection of $n$-Weiss covers as described by Dmitri Pavlov. In the homotopy calculus, we start with a "cover" being any strongly cartesian cube in $Spaces^{op}$; the $n$th localization is obtained by restricting to cubes of dimension $\geq n+1$. There's no Weiss anything in sight.
Here is a nuts-and-bolts comparison, which I think lurks behind the Rezk / Lurie approach to Goodwillie calculus in Higher Algebra, 6.1.1. I'm comparing to 6.2.2 in Higher Topos Theory for the sheaf case.
Recall that a functor $F: \mathcal C \to \mathcal D$ is $n$-excisive if it takes strongly cocartesian $n$-cubes to cartesian $n$-cubes. So what we have are a bunch of "covering" diagrams on $\mathcal C^{op}$ (given by the $n$-cubes which are strongly cocartesian in $\mathcal C$) and we are localizing $Fun(\mathcal (C^{op})^{op}, \mathcal D)$ with respect to these "covers", just like when we localize a category of presheaves at the "covers" of a Grothendieck topology to get sheaves.
So much could be said of any localization, but the analogy extends further, to the way that the localization is computed. For sheafification, we have the Grothendieck plus construction $F^+(C) = \varinjlim_U \varprojlim_{C' \in U} F(C')$ where the colimit is over covers $U$ of $C$. The sheafification of $F$ is computed by iterating this construction [1]. In Goodwillie calculus we do exactly the same thing, but it's simpler because there's always a unique finest cover of any object $C \in \mathcal C$, namely the unique strongly cocartesian cube with $C$ at the initial vertex and the terminal object at all of the vertices adjacent to the terminal vertex. This construction is called $T_n(F)(C) = \varprojlim_{C'} F(C')$ where the limit is over the aforementioned finest cover (the colimit was computed by evaluating at the finest cover). We iterate this construction to produce the $n$th polynomial approximation $P_n(F)$.
If you look at Lurie's treatment, the analogy extends further to the reasoning why this construction correctly computes the localization of $F$. In each case, one shows that the resulting functor is local by inputting a cover $U$ of an object $C$, and then factoring the natural transformation $FU \Rightarrow F^+U$ (resp. $FU \Rightarrow T_n(F)U$) through a limit diagram; applying this factorization to each $F^{+\alpha}U \Rightarrow F^{+\alpha^+}U$ (resp. $T_n^\alpha(F)U \Rightarrow T_n^{\alpha^+}(F)U$) produces a cofinal chain of limit diagrams [2]. Universality comes because the map $F \to F^+$ (resp. $F \to T_n(F)$) is local by construction.
Then the verification that this localization is left exact proceeds in the same way in both cases: the construction $F \mapsto F^+$ (resp. $F \mapsto T_n(F)$) preserves finite limits because (finite) limits commute with limits [3]. Then the localization functor commutes is a filtered colimit of these functors, so it also commutes with finite limits, because finite limits commute with filtered colimits.
[1] In the case of ordinary sheaves, the iteration actually stabilizes after two steps, but in homotopical contexts it may need to be iterated transfinitely.
[2] In the sheaf case, we need to iterate long enough so that the colimit over the $F^{+\alpha}$'s commutes with th limit; in the Goodwillie case, we usually just assume we're in a situation where finite limits commute with filtered colimits, and then since our "covers" are finite, this suffices.
[3] In the sheaf case, we need to know that so do filtered colimits.