It's been argued here on MSE that adding points doesn't make sense (e.g. Ray Toal's answer here: What is the difference between a point and a vector?). The reasoning was that points can be thought of as locations, and vectors can be thought of as displacements. So, adding e.g. Moscow and Los Angeles shouldn't make sense.
But upon considering the fact that the difference between two points is a vector, I came up with the following counterexample. Let $M=[m_1,m_2]$ be the location of Moscow and let $L=[l_1,l_2]$ be the location of Los Angeles. The difference of the locations is a vector, i.e. $$L-M=[l_1,l_2]-[m_1,m_2]=(l_1-m_1,l_2-m_2)$$ (notice the change of brackets). Now, adding $L$ and $M$: $$L+M=L-(-M)=[l_1,l_2]-(-[m_1,m_2])=[l_1,l_2]-[-m_1,-m_2]=(l_1+m_1,l_2+m_2)$$ which clearly is something. Also, ignore Earth's curvature.
So, why do people say that one can subtract two points but not add two points? Do they forget that addition can be turned into subtraction?
I would say that the statement "$L + M = L - (-M)$" is not so much false as it is undefined: when we talk about the plane as a bunch of points then there isn't a natural 'partner point' $-M$ for every point $M$. (Fedja above offers a way out by writing $-M = 0 - M$ but that would make $-M$ a vector rather than a point).
Geometrically speaking there is no 'standard' $-M$ for every $M$. If you want to define it by something like '$-M$ is the point you get by rotating $M$ 180 degrees around the origin' then you run into the trouble the other other two answers are talking about: it singles out one arbitraty point (the origin) in your plane and makes it really important for no particular reason.