It's clear that music is not a language at all, at least not in the way that the things we normally call languages are.
There is no way to say "pass the salt, please", or "That will be fourteen pounds eight shillings and sixpence, but I will accept your finest camel instead" in music. You can set the words to music, of course, but that is cheating.
This is kind of self-evident, but I think that it is necessary to point it out in this context.
Music is, however, a medium of communication, and as such it is capable of saying things which are difficult or impossible to say with words. I can try to describe the feeling I had listening to the opening of Beethoven's C# minor quartet one night several years ago, lying drunk on the carpet, but any combination of words I might come up with will be trite by comparison. You just have to have been there, and what's more you have to have been me. That's just how it is. I'm sure some others have had such moments too.
Perhaps it is apt to say that music communicates not ideas (like real languages), but emotion, in a very pure form. And not only simple individual emotions, but complex palettes of the things can be combined and transmitted, pretty much in parallel, in a way which words are generally too clumsy to achieve.
Not only can music say things which are generally inexpressible through words, but it also can have a certain universality. By this I mean that sometimes we can have a strong affinity for music that we have had no real cultural exposure to. Often, we don't have to learn anything much for the meaning to communicate itself.
A musician friend of mine, an american lady who is a professional classical singer, has had a lifelong obsession with dhrupad (a style of North Indian classical music which sounds quite outlandish to many western-ears). She has not only become very skilled at it, she has performed successfully to large audiences in India, learned the Marathi language, and so on. Although I am quite a skilled musician, that music does not speak to me with the same resonance.
However, the sounds and rhythms of black american jazz impressed themselves on me deeply, even as a child. My earliest clear musical memory is of hearing Louis Armstrong on TV in the late 60's, before I had any idea who he was. This was in a rural area of Wales, and I had no previous exposure to that sound that I know of. That joyful sound stayed with me, and later on I did indeed develop a strong passion for that genre.
Who knows why it is that some types of musical "language" speak so clearly and immediately to some specific people, independently of their cultural or educational background? We don't really know, but the phenomenon does exist. It does not always happen, far from it, and some genres will remain forever opaque to some listeners, even with the best intention to "get it". (On some level, I believe that the answer may eventually found in some deep correspondence between certain types of rhythm, combinations of frequency, and something in our bio-emotional apparatus. But this is pure conjecture, and I don't much care if we never find the answer. Like other good things in life, it just works.)
At any rate - I think that this is what is meant by talk of music as a "universal language". Taken literally, the epithet is cute and inexact of course- but the English language is often like that, and that does not reduce its communicative power. There are many other phrases that are somewhat ridiculous when taken literally, but which do have a useful function.