There is a simple nice fact which holds in the tropical plane that has no counterpart in algebraic geometry (nor in any kind of standard geometry I might think of): two tropical lines always "intersect" in a single point... even if they coincide!

Of course this property relies on the fact that "intersection" is not defined in the usual way. We define the "intersection" of two tropical curves $C_1$ and $C_2$ as follows: the union $C_1\cup C_2$ has a natural cellularization into vertices and edges, and "$C_1\cap C_2$" is the union of the vertices contained in the set-theoretic intersection $C_1\cap C_2$. One may also define a multiplicity on each intersection point. With this definition, the intersection of a tropical curve with itself is the union of its vertices.

Therefore two (possibly coinciding) tropical lines always intersect in a point. By defining anagously an appropriate (dual) notion of "span", the dual sentence is also true: two (possibly coinciding) points always span a single line.