I know the definition of intersection multiplicity in algebraic geometry. However, I think it is possible to define it for some sort of non-algebraic functions such as $y=\sin x$.



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> How to define this?
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Intersection multiplicity in the non-algebraic case
If we work in the category of holomorphic functions, then we can give the following definition, that generalises to the complex-analytic setting the classical intersection multiplicity used in algebraic geometry. 



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> **Definition.** Let $f$, $g$ be two holomorphic functions defined in a neighborhood of a point $\boldsymbol{z\_0} \in \mathbb{C}^2$ and let $$X \colon \{f =0\}, \quad Y \colon \{g=0\}$$ be the corresponding (germs of) analytic varieties. Then the intersection multiplicity of $X$ and $Y$ at the point $\boldsymbol{z\_0}$ is $$m(X, \, Y; \, \boldsymbol{z\_0})= \dim\_{\mathbb{C}} \mathcal{O}\_{\mathbb{C}^2, \,\boldsymbol{z\_0} }/(f, \, g),$$
>  where $\mathcal{O}\_{\mathbb{C}^2, \,\boldsymbol{z\_0} }$ stands for the analytic local ring of $\mathbb{C}^2$ at $\boldsymbol{z\_0}.$
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See for instance 


T. De Jong, G. Pfister: *Local Analytic Geometry: Basic Theory and Applications*, Chapter 5.


