1. Counterexample: The transformation $$Q^1~=~2q^1 ,\qquad P_1~=~2p_1,\qquad Q^2~=~\frac{1}{2}q^2 ,\qquad P_2~=~\frac{1}{2}p_2 $$ preserves phase space volume & orientation, but is not a symplectomorphism.$^1$

2. For 2D phase space, the canonical phase space volume form $$\Omega~=~\frac{1}{n!}\omega^{\wedge n}$$ agree with the symplectic 2-form $\omega$ itself, so that orientation & volume preserving transformations agree with symplectomorphisms.

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$^1$ Here we will assume that OP defines a canonical transformation (CT) as a symplectomorphism. Be aware that several non-equivalent definitions of CTs appear in the literature, cf. e.g. this Phys.SE post.

1. Counterexample: The transformation $$Q^1~=~2q^1 ,\qquad P_1~=~p_1,\qquad Q^2~=~\frac{1}{2}q^2 ,\qquad P_2~=~p_2 $$ preserves phase space volume & orientation, but is not a symplectomorphism.$^1$

2. For 2D phase space, the canonical phase space volume form $$\Omega~=~\frac{1}{n!}\omega^{\wedge n}$$ is the symplectic 2-form $\omega$ itself, so that the orientation & volume preserving transformations are the symplectomorphisms.

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$^1$ Here we will assume that OP defines a canonical transformation (CT) as a symplectomorphism. Be aware that several non-equivalent definitions of CTs appear in the literature, cf. e.g. this Phys.SE post.

Counterexample: The transformation $$Q^1~=~2q^1 ,\qquad P_1~=~2p_1,\qquad Q^2~=~\frac{1}{2}q^2 ,\qquad P_2~=~\frac{1}{2}p_2 $$ preserves phase volume & orientation, but is not a symplectomorphism.

1. Counterexample: The transformation $$Q^1~=~2q^1 ,\qquad P_1~=~2p_1,\qquad Q^2~=~\frac{1}{2}q^2 ,\qquad P_2~=~\frac{1}{2}p_2 $$ preserves phase space volume & orientation, but is not a symplectomorphism.$^1$

2. For 2D phase space, the canonical phase space volume form $$\Omega~=~\frac{1}{n!}\omega^{\wedge n}$$ agree with the symplectic 2-form $\omega$ itself, so that orientation & volume preserving transformations agree with symplectomorphisms.

--

$^1$ Here we will assume that OP defines a canonical transformation (CT) as a symplectomorphism. Be aware that several non-equivalent definitions of CTs appear in the literature, cf. e.g. this Phys.SE post.