CAPM is neither a cross sectional model, nor a time series model!
The classic CAPM formula \begin{equation} \operatorname{E}(R_{i})-R_{0}=\beta_{i}(\operatorname{E}(R_{M})-R_{0})\qquad i=1,2,\cdots,N \label{E:CAPM}% \end{equation} is a relation on expected return, not on return (random variable) itself.
In Econometrics, a cross sectional or a time series model is talking about conditional expectation function, say $$ \operatorname{E}\left( \left. Y\,\right\vert X\right) =\alpha+\beta X $$ or equivalently $$ Y=\alpha+\beta X+\epsilon $$ with $\operatorname{E}\left( \left. \epsilon\,\right\vert X\right) =0$ (for consistent estimators, the mean independence is relaxed to orthogonality $\operatorname{E}(\epsilon X) =0$)
$\operatorname{E}\left( Y\right) $ is a number, but $\operatorname{E}\left( \left. Y\,\right\vert X\right) $ is a random variable: Let $Y=a+bX+\epsilon$ and (joint normal distribution) $$ \begin{bmatrix} X\\ \epsilon \end{bmatrix} \sim\mathrm{N}\left( \begin{bmatrix} \mu_{X}\\ 0 \end{bmatrix} , \begin{bmatrix} \sigma_{X}^{2} & \rho\sigma_{\epsilon}\sigma_{X}\\ \rho\sigma_{\epsilon}\sigma_{X} & \sigma_{\epsilon}^{2}% \end{bmatrix} \right) $$ with $\rho>0$. Then $$\operatorname{E}\left( Y\right) =\mu_{Y}=a+b\mu _{X}=a+b\operatorname{E}\left( X\right) $$ However \begin{align*} \operatorname{E}\left( \left. Y\,\right\vert X\right) & =a+bX+\operatorname{E}\left( \left. \epsilon\,\right\vert X\right) \\ & =a+bX+\left( X-\operatorname{E}\left( X\right) \right) \rho \sigma_{\epsilon}^{\,}/\sigma_{X}^{\,}% \end{align*} Note that OLS estimator is NOT consistent because of endogeneity. Say $\mathrm{cov}\left( \epsilon,X\right) =\rho\sigma_{X}\sigma_{\epsilon}>0$.