The majority of "Liquid Serotonin" is a homeopathic product. Whether it does anything or not probably depends on whether you believe in homeopathy IMO. Homeopathic remedies are prepared by repeatedly diluting a substance in water or alcohol. In true homeopathic remedies, dilution usually continues to the point where no detectable level of the original substance remains. Originally, homeopathy was based on the idea that a small amount of a substance that causes the symptoms in healthy people will treat those symptoms in sick people - this idea in itself is not crazy and is sometimes used in medicine. But what makes homeopathy unusual is the extremely tiny quantities of the substance used, that, as far as we currently understand, are not enough to have any effect. The theory given to explain it is that perhaps the water can capture come sort of vibration, memory or energy of the substance. This is not provable with current science. However, what we do know is that most clinical studies on homeopathic remedies found their effects to be equivalent to (or only very slighter better than) placebo.
The strange thing is that I have seen this "Liquid Serotonin" sold for all sorts of purposes/symptoms, some which you would associate with having too much serotonin and some which you would associate with having not enough serotonin. Even if I did believe in homeopathy, it sounds like it is being misleadingly marketed as a panacea capitalizing on the fact that a lot of people have heard only a little about serotonin and assume serotonin=good.
Some so-called "homeopathic" remedies these days, are not true homeopathic remedies, meaning they have not been quite so heavily diluted, but even if the liquid were to contain pure serotonin, serotonin itself cannot pass through the blood-brain barrier, only its precursors can, so taking serotonin would presumably only have an effect on other parts of your body, such as the digestive system (too much serotonin in the digestive system can cause unpleasant symptoms like diarrhea and nausea). This BBB problem is why serotonin itself is not used in Western medicine and we have so many drugs that indirectly raise serotonin levels or slow its removal from the brain.
If you want to raise your serotonin levels without prescription drugs, there are far better options which have actually been proven to work. For example, taking supplements of serotonin precursors, such as tryptophan or 5-HTP. Note that you should really be taking a balance of other amino acids as well, especially if you don't get enough from your diet, because serotonin precursors need the help of other amino acids to effectively pass the blood brain barrier and you also do not want an imbalance of too much serotonin and not enough of other neurotransmitters such as dopamine (unless you are only using it in situations where you have way too much dopamine), and should also take B vitamins, as they are needed to convert 5-HTP to serotonin. Studies have shown tryptophan or 5-HTP supplementation to be effective even without supplementing these other things, but taking B vitamins and amino acids alongside them makes them more effective and decreases the likelihood of side effects such as a decrease in other neurotransmitters.