No that is not true. If you identify as an asexual it has absolutely nothing to do with what you eat but with how you generally feel sexual attraction towards others. However if you are not sure if you are asexual or not, it might be worth exploring if there could be other reasons like a medical reason behind it. Still, if you have never felt sexual attraction in your life and think the act of sex seems a bit gross, there is a big chance you might be asexual.
They say that chocolate may have a similar effect to the brain than sex (producing the same hormones or something like that), so probably the comic took the idea from that.
About needing to eat sugar, that means that your body is developing a resistence to insulin, so you have insulin for longer un your blood. The insulin tells your cells to consume the sugar, but it also prevents your body to go for your stored fat (using fat would not let the sugar to be burnt), so when the sugar is gone and the insulin is still there your body asks for more sugar, because you can't use your fat and you are low on energy. If you keep eating sugar the resistence to insulin will get stronger.
In the long term it will get harder to maintain and lose weight and in the end you can develope diabetes. So be careful.

Is it true that if you eat too much sweets you will lose your sex drive?
Because I eat sweets every day like crazy (I know it's not healthy, but I just feel strong need to eat them) and I really don't have any interest in having sex, I thought I am asexual, but is it actually because I eat a lot of sweets?