basic definition of vector-fields on a manifold

Question

Many textbooks introduce vector-fields on a manifold $M$ along the lines of $ X = X_i \frac{\partial}{\partial x_i} $ where -without further ado- $\frac{\partial}{\partial x_i}$ is introduced as a basis vector in Tangent space. Given that the same symbol is widely used as a differential operator in mathematics, should I think about $X$ in terms of an operator rather than a vector? Or is that notation a clever suggestive trick to help ease calculations - similar to physicists speaking of "ket-" $ | \psi>$ and "bra-" $ < \Phi | $ vectors in a physical Hilbert space that "magically" turn into a scalar-product $ <\Phi|\Psi> $ when "meeting" in a calculation? Similar conceptual difficulties arise for me when in the the definition of 1-forms $\omega = \omega_1 dx $ the object $dx$ is introduced as a "unit" vector when I am used to thinking about $dx$ as an infinitesimal quantity. Cheers!