It is normal to taste eye drops in the mouth after you drip the eye drops in the eyes, because the eyes are connected to the sinuses by the canalis nasolacrimalis while the nasal cavity is connected with the oral cavity. Therefore, eye drops can flow through the canalis nasolacrimalis into the nasal cavity, and then into the mouth. The taste does not last long, only a few minutes and if you feel the flavor of the eye drops bitter, you can eat a mint to mask the taste.
Yes, the eyes are connected with mouth and nose. The nerves inner the body is interlinked. Many people have the same experience that they have tasted the eye drops in the mouth after they use the eye drops. That is to say, if you have any problems at the throat or nose, like the cold or sinus infection, it will easily cause your eyes problems. You should take care of this. The eyes health is very related with the health with other organs. When you have the cold problems, you'd better use the medical care to stop the infection on the eyes. You should pay special attention during that time on the eyes. At the same time, the eyes drops you use should be in the small amount. Or else, the abundant material will be absorbed by the eyes.
A lot of people, including me, have the same experience to be honest. Truly, our eyes are connected to many parts of our body including our nose, that is important, our mouth, that is your problem, and your ears, liver, etc. In a word, you don't have to worry because that is common and you should thus try to protect your eyes and improve your health so as not to develop eye disorders.