The problem as stated is not well formulated; there are different ways it could be given a precise mathematical meaning, but then the main content of the question becomes the way in which it is made precise.

One thing missing is a specification of the probability measure, or a class of probability measures, for "random integer" and "random polynomial". Also implicitly missing, although not explicitly mentioned, is either a mathematical or a practical model of computation.

If you choose any particular probability distribution on polynomials with integer coefficients, then for any $\epsilon$ there are particular integers $m$ so that the probability of two of the polynomials agreeing at $m$ is less than $\epsilon$. As an example, if you consider any set of polynomials whose coefficients are chosen with any probability measure on integers between -1,000,000 to +1,000,000, then the value for $n = 2,000,001$ is definitive. This may not help you with your actual problem, because computing the value when $n = 2,000,000$ may involve computation with integers of greater than than machine precision, depending on the degree and the machine. If it's a polynomial of degree 10, it's probably better to compute its value on all integers between -5 and 5, rather than one integer of size 2000000; but there are many possible strategies for computation, and this gets into a different set of issues.

For a probability measure that does not have finite support, you can't usually determine identity of the polynomial with absolute certainty from value at a particular integer, but it can still be made as nearly certain as you like.

Similarly, the value on any real number chosen from a distribution with no atoms, or the value on a single known-to-be transcendental real, is definitive --- but computation up to machine precision might or might not tell you equality.