I just wanted to mention another way to recover the basis of holomorphic differentials and the failure of surjectivity given in the comments : an hyperelliptic curve of genus $g\geq 2$ can be expressed as a 2-to-1 cover of $\mathbb{P}^1$ ramified along $2g+2$ points, $f\colon X\to \mathbb{P}^1$. In particular, using double covers formulas we see that
$$ f_*\mathcal{O}_X = \mathcal{O}_{\mathbb{P}^1}\oplus \mathcal{O}_{\mathbb{P}^1}(-g-1) \qquad K_X \cong f^*K_{\mathbb{P}^1}\otimes f^*\mathcal{O}_{\mathbb{P}^1}(g+1) \cong f^*\mathcal{O}_{\mathbb{P}^1}(g-1) $$
so that
$$ H^0(X,K_X) = H^0(\mathbb{P}^1,f_*K_X) = H^0(\mathbb{P}^1,\mathcal{O}_{\mathbb{P}^1}(g-1)\otimes f_*\mathcal{O}_X) = H^0(\mathbb{P}^1,\mathcal{O}_{\mathbb{P}^1}(g-1)) $$
and by the same reasoning
$$ H^0(X,K^2_X) = H^0(\mathbb{P}^1,f_*K^2_X) = H^0(\mathbb{P}^1,\mathcal{O}_{\mathbb{P}^1}(2g-2)\otimes f_*\mathcal{O}_X) = H^0(\mathbb{P}^1,\mathcal{O}_{\mathbb{P}^1}(2g-2)) \oplus H^0(\mathbb{P}^1,\mathcal{O}_{\mathbb{P}^1}(g-3)) $$
Hence, the multiplication map $H^0(X,K_X)^{\otimes 2}\to H^0(X,K^2_X)$ can be interpreted as
$$ H^0(\mathbb{P}^1,\mathcal{O}_{\mathbb{P}^1}(g-1))^{\otimes 2} \to H^0(\mathbb{P}^1,\mathcal{O}_{\mathbb{P}^1}(2g-2)) \oplus H^0(\mathbb{P}^1,\mathcal{O}_{\mathbb{P}^1}(g-3)) $$
which is clearly not surjective as soon as $g\geq 3$, whereas it is surjective for $g=2$. However, one can prove in the same way that the map $H^0(X,K_X)^{\otimes 3} \to H^0(X,K_X^{\otimes 3})$ is not surjective when $g=2$.

An argument for surjectivity can be found in Arbarello-Cornalba-Griffiths-Harris in their discussion of extremal curves.