Would connecting multiple light socket splitters together decrease lumens
First, is it possible?
A 1000 lumen LED light usually draws about 10 W. So you would draw a total of 19 * 10 = 190 W. If you are in a 220-240 V country, that's about 1 A. This is definitely within the allowed power on a single socket (usually 16 A). Consider that you hair drier probably draws 10 times as much.
Next, do you get a total luminous flux equal to the sum of each?
It depends on the shape of the lights. If the bulb emits light in all directions, some of it will be blocked by other bulbs in the path, or by whatever hardware you used to hold all those lights. Some of that will be reflected, some will be absorbed (though unless your bulbs are very peculiar, or you painted everything black, I would expect that to be a very small proportion). If all bulbs emit in the same direction (away from where other bulbs may be), they would indeed add up.
Note how many LED light bulbs are actually composed of a lot of small LEDs. This is quite common practice.
Edit
Note that this answer was written before you added the picture and the mention of 17W LEDs. I thought you were talking about regular power strips, rather than these weird splitters I had never seen. You would need to check the max current they allow, which may be much lower than what a power strip allows. Also take into account the higher power (and thus current) compared to my assumptions.