Adding two velocities which are comparable to the speed of light does not obey vector addition rules. eg:
I am moving at a velocity $c/2$ wrt you along $x$ axis.
I throw an object with $c/2$ velocity wrt me along $x$ axis.
Vector sum of $c/2$ and $c/2$ is $c$.
But the velocity of object wrt you would be less than c.
Forget relativity for a moment.
John and Mary are both standing at the origin, both traveling eastward at speed 1. Let $N$, $E$, $W$ and $S$ be the unit vectors in the four directions. John is facing eastward, using the frame $(E,N)$, and therefore says that his velocity vector is $(1,0)$. Mary is facing north, using the frame $(N,W)$, and therefore says her velocity is $(0,-1)$. If you add these pairs of coordinates together, you get the expression $(1,0)+(0,-1)=(1,-1)$, which represents absolutely nothing of physical (or mathematical) interest.
As a general rule, if you're going to add vectors by adding their coordinates, you'd better first express them both in the same frame. This works fine in classical mechanics as long as John and Mary are both facing east, using the same frame. If they're not, it doesn't work.
Your two relativistic observers are facing different directions in spacetime, expressing their velocity vectors in terms of different frames. It makes perfect sense to add those vectors, but if you want to do that by adding coordinates, you have to first express them both in the same frame.
The (normalized) tangent vectors to John and Mary's worldlines are what you should think of as their velocities in the context of relativity. If you want to add them coordinate-wise, you must first pick a frame in which to express them both. Then you've at least got a meaningful vector sum. Only now can you start thinking about the physical meaning of those coordinates.
