Existence of graph spanners
Strictly speaking it is possible to construct $2k$ spanners but, under some assumptions (see below), their size will never be asymptotically better than the size of the best $2k-1$ spanner.

Indeed, assuming Erdős Girth Conjecture, there are graphs $G=(V,E)$ with girth (length of the shortest cycle) $2k+2$ and $\Omega(n^{1+\frac{1}{k}})$ edges.

A $2k$ spanner $S$ of $G$ cannot exclude any edge $(u,v) \in E$. Indeed, if $(u,v)  \in E \setminus S$, either $u$ and $v$ are disconnected in $(V, S)$, or their distance in $(V, S)$ is at least $2k+1$. This is a contradiction.

This means that any $2k$ spanner of $G$ must have at least $\Omega(n^{1+\frac{1}{k}})$ edges. However, $O(n^{1+\frac{1}{k}})$ edges are already enough to guarantee the existence of $2k-1$ spanners in any graph, as you point out.