I don't think anyone but the designers of the language could possible answer this question. I am not nearly an expert on the subject, but I will provide just my opinion.
By reading the JLS 15.12 about Method Invocation Expressions, it is pretty clear that choosing the right method to execute is an already complicated compile-time process; above all after the introduction of generics.
Now imagine moving all this to the runtime just to support the single feature of mutimethods. To me it sounds like a small feature that adds too much complexity to the language, and probably a feature with certain amount of performance implications now that all these decisions would need to be made, over and over, at runtime, and not just once, as today it is, at compile time.
To all these we could add the fact that due to type erasure it would be impossible to determine the actual type of certain generic types. It appears to me that abandoning the safety of the static type checking is not in the best interest of Java.
At any rate, there are valid alternatives to deal with the multiple dispatch problem, and perhaps these alternatives pretty much justify why it has not been implemented in the language. So, you can use the classical visitor pattern or you can use certain amount of reflection.
There is an outdated MultiJava Project that implemented mutiple dispatch support in Java and there are a couple of other projects out there using reflection to support multimethods in Java: Java Multimethods, Java Multimethods Framework. Perhaps there are even more.
You could also consider an alternative Java-based language which does support multimethods, like Clojure or Groovy.
Also, since C# is a language pretty similar to Java in its general phillosopy, it might be interesting to investigate more on how it supports multimethods and meditate on what would be the implications of offering a similar feature in Java. If you think it's a feature worth having in Java you can even submit a JEP and it may be taken into account for future releases of the Java language.