Number of points at which a tangent touches a curve

My teacher told me that we are mistaken coming out of school that tangent tocuhes a point at one point. According to him, a tangent is just a special type of secant where two points share the same position but actually are just two different points.

My friend said it could happen if we think of tangent as a secant where the two points tend to each other.

I am dissatisfied with it. According to my teacher, the points share the location but are different. Since a point has only its position as its property, how are they differentiated among themselves? E.g. if we take the set of points on which tangent touches the curve (in a sufficiently small neighbourhood around the points), we get a single point since a set doesn't allow redundant entries.

(Moreover, my teacher's tried to explanation his position using quadratic formula taking a quadratic equation as example; but I fail to recognise its relevance except that a quadratic equation has at most two roots, and maybe that tangent has a solution with the curve. But isn't it the problem to begin with, how many solutions do the two of them have?)

And I could agree with my friend too, but if we take a curve to be $y = x^2$ and tangent to be $y = 0$ than there is no point except $(0,0)$ that is common to them. No point in any small neighbourhood around $(0,0)$ is common to both. That seems to only reinforce myself.

Question

How many points do a tangent and its curve actually share? And if they are more than one, how can they be differentiated? That is, what is the application of treating two positions with same position differently?

Literally, your teacher is wrong. The tangent line meets the curve at the one point to which it is tangent, and as you note in your post, this point is one point. (Here I am ignoring the fact that the tangent may also intersect the curve at some other, unrelated point, if the curve is not convex --- this is an unrelated issue.)

What your teacher has in mind, though, is that the curve is a limit of secants through a pair of points, the limit being taken as the two points in the pair tend to the one point at which you are taking the tangent.

Related to this picture of taking a limit of secant lines, in some parts of mathematics, one also says that the tangent meets the curve in a "double point". But this does not mean that the tangent meets the curve in two points; rather, it is a short-hand way of expressing the particular way in which the tangent line meets the curve at the point of tangency.

The fact that the tangent to a curve, defined as the limit of a variable secant, cuts the curve in two indistinguishable points is clear, geometric...and completely nonsensical fron a rigorous point of view!

Grothendieck in the 1950's found the solution to that centuries old charade: his scheme theory defines the intersection as the contact point plus a ring whose size reflects the degree of tangency.
In your example of the tangent $y=0$ to $y=x^2$, the ring is $\mathbb R[x]/(x^2)$ and has size (=dimension as a vector space) 2.
In the case of the tangent $y=0$ to the curve $y=x^3$, the size would be 3, indicating a higher contact of the tangent to the curve.
And of course for the curve $y=x^n$, the size would be n.
One of the revolutionary aspects of this point of view is that tangency becomes a completely static concept: no complicated calculations of limits of secants are involved.

[This answer is at a level more advanced than calculus, but less so than one might think.
It might help put things in perspective and perhaps serve as a slightly enigmatic magnet to a more sophisticated version of geometry.]