ECC goes from "decimal" (actually, real) to integer by reusing the sames equations (for the curve, point addition or doubling), applied to some finite field instead of the infinite field $(\mathbb R,+,*)$. The "curve" becomes a set of finitely many points (as nicely illustrated in that other answer). The intuition that we could keep the geometrical construction does not immediately materialize, especially for point doubling.
Update: but not all hope is lost. The last picture in this tutorial shows that, when the finite group is $(\mathbb Z_p,+,*)$, we can make a geometric construction of point addition $A+B$ when $A\ne B$. And for doubling point $A$ we can take another point $B$ and say that $2\times A=2\times A+(B-B)=(A+B)+(A-B)$ and compute the later geometrically with three point additions.
Note: Just restricting from $\mathbb R$ (reals) to the subset $\mathbb Z$ (integers) leads nowhere, because so few points on the curve fall onto integer coordinates.
The geometric construction in the question applies on the Cartesian plane. It lets one establish the formulas for the point addition operation, and its special case doubling, as equations between Cartesian coordinates in the field $(\mathbb R,+,*)$. That point addition is an operation on the points of curve; it is internal (result is on the curve), and commutative. We can include an additional point as identity element, and define that elements have an additive inverse by symmetry relative to the horizontal axis; all that is geometrically evident and consistent. Then algebra based on the formulas shows the surprising fact that the point addition operation is associative (reportedly, there's also a non-trivial geometrical proof; what follows is an illustration only). We have a group!

Credit: Thomas Cooper, who created a GNU Octave script to generate this file; Public Domain.
Elliptic curve cryptography replaces $(\mathbb R,+,*)$ with a finite field, perhaps $(\mathbb Z_p,+,*)$. This is not performed by considering points of the curve that fall on integer points, because there are very few; for example $y^2=x^3−2x+1$ has only 3 integer solutions: $(x,y)\in\{(0,1),(0,-1),(1,0)\}$.
Rather, replacing $(\mathbb R,+,*)$ with a finite field is made by using the curve's equation in that finite field, as well as the formulas established for point addition and doubling; that yields an operation that is commutative, associative, with identity element, and an opposite for any element; again, the axioms for a group.
The resulting "curve" looks like an haphazard set of points (with symmetry along an axis inherited from that of the original curve), as nicely illustrated in that other answer. The geometric (or at least, visual) aspect of the construction vanishes, except for opposites. In particular, we can no longer visually define a tangent, or the intersection of the line joining two points of the "curve" with a third point of the "curve". And what used to be a "slope" now involves a multiplicative inverse in the finite field, which computation is performed radically differently (e.g. by the Extended Euclidean algorithm or by exponentiation) from computing the multiplicative inverse in $\mathbb R$.
Why are you guys so afraid of the word "division"?
Because for a neophyte, this is likely to be taken as a division as in $\mathbb{R}$. $\endgroup$