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](https://ncatlab.org/nlab/show/Thom+spectrum) and [G-structure](https://ncatlab.org/nlab/show/G-structure#InTermsOfBfStructures).