When the professor says that x is a "meaningless symbol" what exactly is meant by that?
It just means that you can think of a polynomial "a + b x + c x^2 + ... + d x^n" as the ordered tuple (a,b,c,...,d), and the "x" and "+" and exponentiation notation are not part of the polynomial but merely there to give suggestive notion. The coefficients of the polynomial are then simply {a,b,c,...,d}, and you can easily define addition and multiplication of polynomials with coefficients from the same ring. By this definition, the polynomial itself does not even include which symbol is used as the indeterminate, although we use "x" in describing the polynomial. For example we have that the product of (1,1) and (1,-1) is (1,0,-1). This is conventionally described by saying that (1+x) * (1-x) = 1-x^2.
But the above is not the only definition of polynomials in use. In algebra, it is common to talk about polynomials as if they are the actual strings of symbols written down including the 'powers' of "x" and "+". There are some subtle errors in reasoning that can creep in when one uses such 'intuitive' notions of polynomials. And unsurprisingly, it will be difficult to properly formalize this notion without going back to the earlier definition. For example, we can 'imagine' that "1 + x y + x^2 + y^3" is a polynomial in both "x" and "y", but is it a polynomial in "x" alone or "y" alone?
Note that the formal definition of a univariate polynomial as a tuple can be extended to multivariate polynomials, by defining a k-variate power series over R as a function from N^k to R, and then defining a k-variate polynomial over R as a k-variate power series that is zero except on finitely many input index-tuples. Note that a bivariate polynomial over R can be viewed as a univariate polynomial over univariate polynomials over R, in the same way that functions can be curried.
Does it imply that the symbol x has no 'meta-lingual' interpretation?
So your professor's answer is actually not precise enough to determine whether he is using the purely rigorous definition. There is a third possible definition, where a polynomial is a function of the form ( x ↦ a + b * x + c * x^2 + ... + d * x^n ), where "+" and "*" can be overloaded and "x^k" is just a short-form for the k-fold product of "x". Such a function will be undefined or crash if given an input x for which any of the multiplications or additions are not defined. In this sense "x" is not really meaningless but is in fact a parameter name. Of course, one can use a different parameter name (as long as it is not used elsewhere) without changing the meaning. The disadvantage of this approach is that it is very difficult to formalize with all the overloading, but it actually corresponds quite closely with how we informally reason.
It is actually not too hard to use the formal definition to capture this notion, via the evaluation map on univariate polynomials over R (where R is a ring), that maps each such polynomial p and object x in R to the value ( a + b * x + c * x^2 + ... + d * x^n ). Similarly for multivariate polynomials. The only thing is that we always have to use this evaluation map to 'apply' a polynomial to some object. People naturally create the short-hand "p(x)" for the above value, despite this conflicting with the formal definition! (By the formal definition, p(1) would be the coefficient of the linear term, but by the convenient short-hand p(1) would be the sum of all the coefficients!)
Does it make any sense to say that "symbols are independent of meaning or interpretation"?
As I hope the rest of my answer has shown, it does not actually have much to do with polynomials and your professor's statements. That said, indeed it is reasonable to say that symbols are just symbols and devoid of meaning unless you interpret them. In mathematical logic we even specifically define an interpretation to be a function on strings of symbols, which is intended to be a map to their 'meanings'.
Can there exist a symbols which doesn't have any "interpretation" or "meaning"? In other words, what is the ontological status of symbols?
Choose an arbitrary collection of 100 pixels in a 100*100 square, and that collection forms a symbol. What meaning does it have? Well, you could tell people that you want to use it to denote something, in which case it has meaning to whoever accepts your decision. Likewise we can say that the answer to your question is "No." because given any symbol I can choose to interpret it to mean whatever I like, and it will have that meaning to me.
Indeed, it is useless to ask whether a string of symbols has meaning (in itself), simply because any meaning can be given to it. What makes a particular interpretation useful is when it can be applied uniformly to a whole collection of strings (such as first-order semantics over a first-order structure applied to first-order sentences) and has non-trivial properties (like soundness and completeness).
