My intuition, for what it's worth, works more easily with the binary expansion of the numbers than their interpretation as physical quantities.

From that perspective, "X equals exactly pi" would normally be assigned finite probability because pi is a computable number with finite Kolmogorov complexity; there is a nonzero chance that two processes will generate the same infinite but computable bit stream.

But "X equals exactly Y" where Y is a random incomputable number, is indeed infinitely improbable, because it amounts to a statement that infinitely many coin flips will come out a particular way; the probability is 0.5^infinity, which clearly converges to zero.