In this paper which surveys the Goodwillie Calculus, the following notation is used in the introduction:
$$\pi_*(P_1 I(X)) \cong \pi_*^s(X)$$
where $X$ is a based space, $I$ is the identity functor, and $P_1$ is a $1$-excisive functor. Does this notation mean that all homotopy groups of $P_1I(X)$ are isomorphic to the stable homotopy groups of $X$, where we replace $X$ by an associated spectrum? If so, is this associated spectrum the suspension spectrum?
As the paper states $P_1(I(X)) \simeq \Omega^\infty \Sigma^\infty X$, and the homotopy groups of the latter are $\pi^s_*(X)$ since the map $\Omega^n \Sigma^n X \rightarrow \Omega^{n+1} \Sigma^{n+1} X $ on homotopy groups (up to shifting) is the stabilization map $\pi_{*}(\Sigma^ n X) \rightarrow \pi_{*+1}(\Sigma^{n+1} X)$. Here $P_1$ is not any 1-excisive functor, but the first polynomial approximation functor, i.e. the first polynomial approximation of the identity on spaces is stable homotopy.