Can two 'different' vector spaces have the same vector?

Strictly speaking, $(v_1,\ldots,v_m,0)\ne (v_1,\ldots,v_m)$. Then again, the obvious injective map $\iota\colon\Bbb R^m\to \Bbb R^{m+1}$ is sometimes viewed as being so natural that we sometimes identify the image $\iota(\Bbb R^m)$ and $\Bbb R^m$ and say, for example $\Bbb R^m\subseteq \Bbb R^{m+1}$ (instead of more correctly $\iota(\Bbb R^m)\subseteq \Bbb R^{m+1}$). In such a context, we would also identify $(v_1,\ldots,v_m)$ with $\iota(v_1,\ldots,v_m)=(v_1,\ldots,v_m,0)$ and say they are equal.

There are contexts where this comes really natural, such as when whe construct, e.g., rationals as equivalence classes of pairs of integers and yet view the integers as subset of the rationals; every step of $\Bbb N\subset\Bbb Z\subset \Bbb Q\subset \Bbb R\subset \Bbb C$ is of this kind: We construct the larger object from the smaller and identify the original set with its different-looking copy in the larger set. However, for the case at hand $\Bbb R^m\to\Bbb R^{m+1}$, the embedding may be what first comes to mind - but it is not really natural. Hence here I would rather not make the identification (unless after specifically mentioning the choice of embedding).

As a sidenote, Let $V$ be any vector space, $v\in V$ any element, and $\alpha$ any object with $\alpha\notin V$ (for example, my neighbour's dog). Then we can define a vector space $V'$ containing $\alpha$ in place of $v$. This way, we can easily construct vector spaces having arbitrary elements in common (and in fact the same elements playing different "roles" in the different vector spaces)

The presumption that two different vector spaces can contain the same vector is correct. For instance, if we consider distinct, though overlapping, subspaces of the same vector space then they certainly contain a common vector. For instance in $\mathbb{R}^3$ the $xy$-plane and the $x$-axis have all their vectors in common, yet they are distinct subspaces. I suppose it depends on what conditions you would like your subspaces to satisfy.

In your example, I would not say the vectors are the same since the space in which we view the vector is important. I would argue instead that

$$ v_2 \;\; =\;\; \left [ \begin{array}{cc} v_1 & 0 \\ \end{array} \right ]. $$

They are inherently different since they are objects of different dimensions, even though they contain "nearly" the same information.