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The present work is focused on the investigation of the existence of compact structures describing anisotropic matter distributions within the framework of modified gravity theories, specifically $f(R,\mathcal{T}$) gravity theory. Additionally, we have taken $f(R,\mathcal{T})$ as a linear function of the Ricci scalar $R$ and the trace of the energy-momentum tensor $\mathcal{T}$ as $f(R,\mathcal{T})=R+2\ensuremath{\chi}\mathcal{T}$, where $\ensuremath{\chi}$ is a dimensionless coupling parameter, and the Lagrangian matter ${\mathcal{L}}_{m}=\ensuremath{-}\frac{1}{3}(2{p}_{t}+{p}_{r})$, to describe the complete set of field equations for the anisotropic matter distribution. We follow the embedding class I procedure using the Eisland condition to obtain a full space-time description inside the stellar configuration. Once the space-time geometry is specified, we determine the complete solution of modified Einstein equations by using the MIT bag model equation of state ${p}_{r}=\frac{1}{3}(\ensuremath{\rho}\ensuremath{-}4B)$ that describes the strange quark matter (SQM) distribution inside the stellar system, where $B$ denotes a bag constant. The physical validity of our anisotropic solution is confirmed by executing several physical tests. It is worth mentioning that with the help of the observed mass values for the various strange star candidates, we have predicted the exact radii by taking different values for $\ensuremath{\chi}$ and $B$. These predicted radii show a monotonic decreasing nature as the parameter $\ensuremath{\chi}$ is moved from $\ensuremath{-}0.8$ to 0.8 progressively. In this case, our anisotropic stellar system becomes more massive and transforms into more dense compact stars. We also perform a detailed graphical analysis of the compact star. As a result, for $\ensuremath{\chi}l0$, the current modified $f(R,\mathcal{T})$ gravity seems promising to explain the observed massive compact astrophysical objects, similar to magnetars, massive pulsars, and Chandrasekhar super white dwarfs, which are not justified in the framework of general relativity. Finally, we note that when $\ensuremath{\chi}=0$, general relativity results for anisotropic matter distributions are recovered.