Given a riemannian manifold $(N,g^N)$ and a submanifold $M\subset N$ the induced metric tensor is definied by $$(h^M)_{ij}(p) = (\partial_i)^k(g^N)_{kl}(\partial_j)^l $$ with some basis $(\partial_i)$, $i=1,...,dim(M)$ of the tangent space $T_pM$. The tangent vectors are identified by the isomorphism of the inclusion map $i: M \to N$ on the tangents spaces $di_p: T_pM\to T_pN$ for some $p\in M$.
Since the tangent space $T_pM$ is a vector space it is isomorphic to some $\mathbb{R}^m$ and I'm free to choose some basis for $T_pM$. But if I do so this basis can be independent on $p\in M$ which implies that $h^M(p)$ is indipendet on $p\in M$.
It seems that I'm not free to choose a basis? But why not?
Even though the tangent spaces $T_p M$ are all isomorphic to $\mathbb{R}^m$, this is not a single isomorphism, but a family of isomorhisms (because their domains are different sets $T_p M$), so your bases do depend on point $p$. This is actually encoded in the notation that you use (the subscript $p$).
Notice that there is no canonical way to identify $T_p M$'s, but if you make such an identification, you will obtain an additional structure on $M$ (known as a connection).