Manifolds whose tangent spaces have a special behavior

Consider an $$n$$-dimensional complex manifold $$M\subset\mathbb{C}^N$$ and let

$$f:\mathcal{U}\subset\mathbb{C}^n\rightarrow \mathcal{V}\subset M\subset\mathbb{C}^N$$

be a local parametrization of $$M$$.

Assume that for all $$p\in\mathcal{U}$$ we have that $$f(p)$$ is a linear combination of $$\frac{\partial f}{\partial x_1}(p),\dots, \frac{\partial f}{\partial x_n}(p)$$.

Is then $$M\subset\mathbb{C}^N$$ an affine subspace of $$\mathbb{C}^N$$?

In general, does there exist a term for a point $$q = f(p)\in M$$ such that $$f(p)$$ is a linear combination of $$\frac{\partial f}{\partial x_1}(p),\dots, \frac{\partial f}{\partial x_n}(p)$$?

The assumption you made is equivalent to stating that for every $$q\in M$$ that the vector $$q\in T_qM$$.
This is satisfied whenever $$M$$ is a portion of a cone, which need not be affine.