Why should $\mathbb Z_p[[T]] \otimes_{\mathbb Z_p} \mathbb Q_p$ be isomorphic to the bounded sequences with values in $\mathbb Q_p$?
The fact is that the tensor product is on $\mathbb Z_p$, so it is not true that $\mathbb Z_p[[T]] \otimes_{\mathbb Z_p} \mathbb Q_p$ is $\mathbb Q_p[[T]]$. Also a priori I can't see that the sequence is bounded.
Consider the map $$\mathbb Z_p[[T]]\otimes_{\mathbb Z_p}\mathbb Q_p\to\{\text{bounded sequences in }\mathbb Q_p\}\\\sum_{i=0}^\infty a_iT^i\otimes z\mapsto(a_0z,a_1z,\ldots).$$
It is clear that the map is well defined, and takes an element of the domain to a bounded sequence, since $z\in\mathbb Q_p$ has a bounded denominator.
Moreover it has inverse given by $$(x_0,x_1,\ldots)\mapsto\sum_{i=0}^\infty p^kx_iT^i\otimes p^{-k}$$ where $p^k$ is the supremum of the absolute values of the $x_i$.
Hence it is an isomorphism.