First use GAGA to pass to complex manifolds. Then use base change to pass to a real manifold....Now, given any smooth embedding (use the Whitney embedding theorem) of your manifold in Euclidean space, assume that the tangent surface at p exists and does everything you want it to. The one-to-one correspondence you ask for, which was not given by the other answer, is:
Given a vector starting at p and lying in the tangent surface, consider the geodesic line lying in your manifold, passing through p, and whose tangent vector in the usual Euclidean sense is the given vector. Problem: to define a derivation of the space of germs of functions on your manifold at p. Hint: First extend your function to the entire ambient space, smoothly. Then use the ordinary directional derivative.
Remark: this is not canonical, but you didn't ask for a canonical isomorphism.
Second remark: you didn't specify whether the field has characteristic zero or not...I assume it does. Third remark: what you wrote down isn't actually the definition of the Zariski tangent space. The definition of the Zariski tangent space is $m_p/m_p^2$ where $m_p$ is the ideal of all algebraic functions which vanish at p.