Let $(M,g)$ be a (smooth) compact Riemannian manifold with boundary. We glue the two copy of $M$, to make it closed Riemannian $M'$.
Q 1.Can we make a modify $g$ near $\partial M$, such that there is a smooth metric $g'$ restricted on either copy, it coincides with the modified metric$g$.
I guess we can do it by product some (smooth)positive-function.
PS: The earlier mistake is pointed by user10354138. Thanks.
In general, no you can't, because the metric need not be smoothly compatible at the boundary, akin to $f(x)=|x|$ is not differentiable at 0, despite its restriction to both $(-\infty,0]$ and $[0,\infty)$ are analytic.
What you could do is to allow a little bit of room to smooth it out, so the restriction is $g$ away from a small tubular neighbourhood of the (original) boundary.