I think that both of your examples, $M_2Spin$ and $M_2O$, arise naturally in the context of Thom spectra induced by $(B,f)$-structures. Given a $(B,f)$-structure $\mathcal{B}= \{f_n: B_n \to BO(n)\}$, the associated Thom spectrum $M\mathcal{B}$ is defined componentwise as: $$ M\mathcal{B}_k = Thom(f_k^*V_k\to B_k), $$ where the maps $\Sigma M\mathcal{B}_k \to M\mathcal{B}_{k+1}$ are given by looking at the pullback square $\require{AMScd}$ $$ \begin{CD} \mathbb{R}\oplus f_k^*V_k @>>> f_{k+1}^*V_{k+1}\\ @VVV @VVV \\ B_k @>>> B_{k+1}. \end{CD} $$ One can 'double' this construction replacing the maps $f_k$ with $\tilde{f_k}$ defined to be the composition: $$ \tilde{B_{2k}}:=B_k\overset{f_k}{\to} BO(k) \overset{\Delta}{\to} BO(k) \times BO(k) \overset{j_{k,k}}{\to} BO(2k) $$ and get a $S^2$-$(B,f)$-structure, a $(B,f)$-structure indexed only on even natural numbers, denoted by $2\mathcal{B}$. By definition, the Thom spectrum $M_2\mathcal{B}$ associated to this new $S^2$-$(B,f)$-structure, is

$$ (M_2\mathcal{B})_{2k} = Thom(\tilde{f_k^*}V_{2k}\to \tilde{B_{2k}}) = Thom(f_k^*V_k\oplus f_k^*V_k \to B_k) $$ $$ (M_2\mathcal{B})_{2k+1} = \Sigma(M_2\mathcal{B})_{2k}. $$ In your case, $M_2Spin$ and $M_2O$, are the Thom spectra associated to the 'doubled' $(B,f)$-structures that classically define $MSpin$ and $MO$, i.e. the $(B,f)$-structures respectively given by the maps $BSpin(k)\to BO(k)$ and ${{\rm id}}_{BO(k)}$.

I think that both of your examples, $M_2Spin$ and $M_2O$, arise naturally in the context of Thom spectra induced by $(B,f)$-structures. Given a $(B,f)$-structure $\mathcal{B}= \{f_n: B_n \to BO(n)\}$, the associated Thom spectrum $M\mathcal{B}$ is defined componentwise as: $$ M\mathcal{B}_k = Thom(f_k^*V_k\to B_k), $$ where the maps $\Sigma M\mathcal{B}_k \to M\mathcal{B}_{k+1}$ are given by looking at the pullback square $\require{AMScd}$ $$ \begin{CD} \mathbb{R}\oplus f_k^*V_k @>>> f_{k+1}^*V_{k+1}\\ @VVV @VVV \\ B_k @>>> B_{k+1}. \end{CD} $$ One can 'double' this construction replacing the maps $f_k$ with $\tilde{f_k}$ defined to be the composition: $$ \tilde{B_{2k}}:=B_k\overset{f_k}{\to} BO(k) \overset{\Delta}{\to} BO(k) \times BO(k) \overset{j_{k,k}}{\to} BO(2k) $$ and get a $S^2$-$(B,f)$-structure, a $(B,f)$-structure indexed only on even natural numbers, denoted by $2\mathcal{B}$. By definition, the Thom spectrum $M_2\mathcal{B}$ associated to this new $S^2$-$(B,f)$-structure, is

$$ (M_2\mathcal{B})_{2k} = Thom(\tilde{f_k^*}V_{2k}\to \tilde{B_{2k}}) = Thom(f_k^*V_k\oplus f_k^*V_k \to B_k) $$ $$ (M_2\mathcal{B})_{2k+1} = \Sigma(M_2\mathcal{B})_{2k}. $$ In your case, $M_2Spin$ and $M_2O$, are (as sequential spectra) the Thom spectra associated to the 'doubled' $(B,f)$-structures that classically define $MSpin$ and $MO$, i.e. the $(B,f)$-structures respectively given by the maps $BSpin(k)\to BO(k)$ and ${{\rm id}}_{BO(k)}$.

All the details and references can be found in the nlab pages Thom spectrum and G-structure.