(A comment to Tyler's answer.) Strictifying pairings from the stable homotopy category to spectra can be tricky. To even get started with an inductive approach let me assume $E$ is connective, so that $E = \tau_{\ge0}E$. Let $p_n : \tau_{\ge n} E \to \tau_{\ge n-1} E$ be the maps in the Whitehead tower, and let $\mu = \mu_{0,0} : E \wedge E \to E$ be the given pairing. The composite $\mu_{0,0} (p_1 \wedge 1) : \tau_{\ge1} E \wedge E \to E$ factors up to homotopy though $p_1$. We may assume that each $p_n$ is a fibration (and that everything in sight is cofibrant), so by the homotopy lifting property there is also a strict factorization as $p_1 \mu_{1,0}$. Hence we can choose lifts $\mu_{1,0} : \tau_{\ge1} E \wedge E \to \tau_{\ge1} E$ and $\mu_{0,1} : E \wedge \tau_{\ge1} E \to \tau_{\ge1} E$ such that $p_1 \mu_{1,0} = \mu_{0,0} (p_1 \wedge 1)$ and $p_1 \mu_{0,1} = \mu_{0,0} (1 \wedge p_1)$. The composites $\mu_{1,0} (1 \wedge p_1) : \tau_{\ge1} E \wedge \tau_{\ge1} E \to \tau_{\ge1} E$ and $\mu_{0,1} (p_1 \wedge 1) : \tau_{\ge1} E \wedge \tau_{\ge1} E \to \tau_{\ge1} E$ agree when projected to $E$, but without further work they may not agree as maps to $\tau_{\ge1} E$. In particular, they may not have a common factorization as $p_2 \mu_{1,1}$ for some $\mu_{1,1} : \tau_{\ge1} E \wedge \tau_{\ge1} E$. So strictifying a pairing is not just a matter of obstruction theory or essential uniqueness of lifts. One may also need to change the models for the spectra involved.

In the case of spectra formed from simplicial sets, there may be a sufficiently functorial (and monoidal) construction of Whitehead towers of simplicial sets, hence also of symmetric spectra in simplicial sets, to ensure that $\mu$ induces compatible $\mu_{m,n} : \tau_{\ge m}E \wedge \tau_{\ge n}E \to \tau_{\ge m+n}E$ for all integers $m$ and $n$, but I do not recall checking this carefully, and if correct, it would be badly model-dependent.

There is a $2$-categorical approach that works. We can choose maps $\mu_{m,n} : \tau_{\ge m} E \wedge \tau_{\ge n} E \to \tau_{\ge m+n} E$, "horizontal" homotopies $h_{m,n} : \mu_{m-1,n} (p_m \wedge 1) \simeq p_{m+n} \mu_{m,n}$ and "vertical" homotopies $v_{m,n} : \mu_{m,n-1} (1 \wedge p_n) \simeq p_{m+n} \mu_{m+n}$. In the case of a Whitehead tower, one can find a $2$-homotopy between the composite homotopies $v_{m-1,n} h_{m,n}$ and $h_{m,n-1} v_{m,n}$, and this suffices to get a pairing of Cartan-Eilenberg systems, hence also a pairing of spectral sequences. These $1$- and $2$-homotopies produce a strict pairing of filtered spectra $Tel(E) \wedge Tel(E) \to E$, where $Tel$ denotes the mapping telescope.

This can surely be promoted to an $\infty$-categorical statement, but the $2$-categorical one suffices for pairings of spectral sequences.