There are two definitions of intersection multiplicity of two complex algebraic plane curves. One of them is given in *An introduction to algebraic curves* by **Griffiths**. Let $t \mapsto (t^k, y(t) )$ be the local normalization of $V_f$ then $(V_g \cdot V_f)_P = ord_P(g(t^k, y(t)))$. The other one is given in *Algebraic Geometry* by **Hartshorne**. $(V_g \cdot V_f)_P = \mathrm{length}(O_p/(f,g))$ as $O_P$ module.

Do these definitions coincide?

There may be some approaches. Showing that completion preserves length may help cause you can thereby catch the normalization in complex analysis to algebraic one. Showing that $\mathrm{length}(O_P/(g, f_1 f_2))= \mathrm{length}(O_P/(g, f_1))+\mathrm{length}(O_P/(g, f_2))$ is also important.