In Volume 1 of Spivak, the Poincaré isomorphism for a connected oriented n-manifold
$PD: H^k(M) \to H_c^{n-k}(M)^*$
is given by
$PD(\alpha)(\beta) = \alpha \cup \beta \in H_c^{n}(M) \cong \mathbb{R}.$
Where the latter isomorphism comes from taking the unique member $\eta \in H_c^{n}(M)$ satisfying
$\int_{M,\mu} \eta = 1$
to 1 (where $\mu$ is the orientation of $M$).
Spivak first constructs Thom classes for an orientable $k$-bundle $\xi: E \to M$ by claiming that there is a unique $U \in H_c^{k}(E)$ with
$\pi^*\eta \cup U = \nu \in H_c^{n+k}(E)$,
where $\eta$ is the compact orientation class on $M$, $\pi$ is the fibration, and $\nu$ is the compact orientation class on $E$.
This appears to me like choosing a vector from a non one-dimensional vector space to attain a value under a given functional, and I do not see the uniqueness or how this follows from the Poincaré Duality Theorem.