I think a mainstream answer would be that symplectic geometry has two (seemingly opposiate, but actually related) aspects: rigidity and flexibility, it is the rigidity aspect that makes symplectic geometry a kind of geometry.
The study of the rigidity of symplectic manifolds dates back to Gromov's groundbreaking work on $J$-holomorphic curves, where $J$ is an almost complex structure on the symplectic manifold $(M,\omega)$ tamed by $\omega$. Using this theory Gromov deduced a lot of interesting facts such as the non-squeezing theorem and the non-existence of simply-connected closed Lagrangian submanifolds in $\mathbb{C}^n$. So unlike topology, the appearance of a non-degenerate 2-form does impose restrictions when studying problems related to embedding, immersion, homotopy or isotopy.
The algebraic nature of symplectic geometry also originates from the existence and abundance of $\omega$-tame almost complex structures, since they have led to constructions of various algebraic structures associated to symplectic/contact manifolds like the quantum cohomology (there is an $E_2$-algebra structure on the chain level), Fukaya category (which is an $A_\infty$-category), linearized contact homology (there is an $L_\infty$-structure on the chain level)... In fact, the algebraic nature of symplectic geometry is usually the reason why many of the rigidity results should hold. For example, a lot of restrictions of Lagrangian embedding can be deduced from the classification of the objects in the Fukaya category, and this method is extremely effective, say, when the quantum cohomology is semisimple as a ring.
On the other hand, since Gromov's theory deals only with $J$-holomorphic curves with finite energy, so you can think of them as sort of more flexible analogues of algebraic curves. This shows that when studying its rigidity aspects, symplectic geometry behaves in some sense more like algebraic geometry. For example, it is a result due to Kollar, Tian, Starr, Ruan, etc. that uniruledness and rational connectedness of smooth projective varieties are invariant under symplectic deformations. Many invariants in algebraic geometry or singularity theory, say the log Kodaira dimension of a quasi-projective variety, or the minimal discrepancy of an isolated singularity can also in some sense be interpreted as symplectic/contact invariants.
Finally, I'd like to mention that when looking at symplectic manifolds from a more flexible perspective, symplectic geometry behaves more like differential topology. This is the reason why symplectic geometry is sometimes referred to as symplectic topology. Parallel to the theory of $J$-holomorphic curves, which treats symplectic manifolds as generalizations of algebraic varieties, when taking a handlebody decomposition perspective, one would naturally expect to relate the geometry of an open symplectic manifold with $H^k(M;\mathbb{Z})=0$ for $k>\frac{1}{2}\dim(M)$ to that of a Stein manifold, that's why Gromov and Eliashberg's h-principle plays a pivotal role there.