Perfect risk sharing in Arrow-Debreu w/ the same subjective beliefs over states
In two details you seem to be mistaken:


You need strict concavity of $v(\cdot)$, not concavity.
According to the definition of quasiconcavity, the function $x^2$ is not quasiconcave, it is quasiconvex.


Your main point is correct. If both functions are linear, then $x_i^1 = x_i^2$ no longer needs to hold, as the goods $x_i^1$ and $x_i^2$
are perfect substitutes for consumer $i$. Hence all consumption decisions are correct as long as $i$ trades at a price ratio of 1. This is indeed the equilibrium price ratio. The same can be said for the other consumer.