Electronic – Am I having a power supply issue with the LED lights strips in series, or have I done somethine else wrong

These strips are usually arranged in numerous parallel segments of a few series LEDs. The RGB strips can be a little different, but they work basically the same way. It looks like your strip is analog control and not digital (which would have little ICs in the strip to control individual LED color), so the specific color ground lines are all in parallel. You should be able to connect numerous strips in parallel - 12V to 12V, red to red, green to green, and blue to blue to blue. That is the same reason you can cut the strips into smaller segments if you want.

There is a limit to how much current can flow through the strip itself, which varies between manufacturers. Your other limit to how many strips can be connected together is how much power your supply / remote controller can handle. It should specify a limit in watts / current or total number of LEDs or parallel segments. That information should be on a sticker on the power supply or be listed in the manual.

From your product page:

Power 72 W

Which is 6A at 12V DC. That means your power supply/controller has to be able to handle at least 144 W (12A @ 12V DC) to connect two strips in parallel. Although, it is never a good idea to run something long term near its maximum ratings, so the supply should really be rated for something closer to 200 W.

If you are able to power multiple strips from your supply, it would be best to power them with a star topology so the current for each strip is separated. For example, connecting Christmas lights together end to end is daisy chaining, and the current for every light strand has to flow through the strands before it. Powering multiple light strands from one power strip is more of a star topology, with the current to each strip only flowing through itself (think of the power strip as your power supply).

At full white you are looking at 30 amps 12V. I doubt your supply handles 30 amps on the 12v rail alone. And that would fry your controller.

Look into the rgb power amplifiers or power injectors. These take the signal from the controller and provide separate control for another strip.

Your text does not mention the power amplifiers. But the diagram does. As long as your supply can safely do 360 watts, you should be good. If anything, you might want to power the strips from the middle instead of the end, to ensure better uniformity in color. But you have it handled.