Let $A,B$ be vector spaces such that $A\subseteq B$. Is it true that $A$ is a subspace of $B$?
I claim that the answer is no, because it is possible that $A$ and $B$ might be equipped with different vector space operations. But what would be a concrete counterexample?
Let $\varphi$ be a non-linear bijection $\mathbb{R} \to \mathbb{R}$. Define vector addition by $$v + w := \varphi(\varphi^{-1}(v)+\varphi^{-1}(w))$$ and scalar multiplication by $$\alpha\cdot v := \varphi(\alpha\cdot\varphi^{-1}(v)).$$ The identity element of addition is $\varphi(0)$ and the additive inverse of $v$ is $\varphi(\varphi^{-1}(0)-\varphi^{-1}(v))$. Then $\mathbb{R}$ is a vector space under these non-standard operations.
Now $\mathbb{R} \subseteq \mathbb{R}$ (where the first copy of $\mathbb{R}$ has the non-standard vector space operations, and the second copy has the standard vector space operations) but $\mathbb{R}$ is not a subspace of $\mathbb{R}$.
Let me give an example of the non-standard vector space operations. Using the non-linear bijection $\varphi : \mathbb{R} \to \mathbb{R}$, $\varphi(x) = x+1$, we see that $v + w := v + w - 1$ and $\alpha\cdot v := \alpha v - \alpha + 1$.