Comment by generationP

These examples miss the mark somewhat. The "2" in "2π" can mean several things (the nonnegative integer 2, the integer 2, the rational 2, the real 2) that are all commonly identified but are different. The "2" in "x^2" usually means the nonnegative integer 2. The "2" in "2x" can usually mean the nonnegative integer or the integer 2, but also the other 2's depending on what x is. The problem is not that the meaning of 2 varies across different expressions, but that it can vary within each single expression.

The best example is perhaps the polynomial ring R[x][y], which consists of polynomials in the variable y over the ring of polynomials in the variable x over the real numbers. Any algebraist would tell you that it is obviously just the two-variable polynomial ring R[x, y] in disguise, because you can factor out all the y-powers and then the coefficients will be polynomials in x. But the rings are very much not the same at the level of implementation, and every time you use their "equality" (canonical isomorphy), you need to keep the actual conversion map (the isomorphism) in the back of your mind.