We agree that in Cantor's diagonal argument, applied to real numbers, the numbers are represented and identified solely by their digits. No further information is available.
We assume that a real number p can be distinguished from a set Q of real numbers q by general considerations, for instance, if p is a transcendental number and Q consists of rational numbers q only.
Of course it would be impossible to distinguish p from all q, because for every digit d_n of p, there is a number q that shares all digits up to d_n with p.
It is possible to construct the complete binary tree from the numbers in Q. That means, it is possible to construct all possible digit sequences from the numbers in Q. It is also possible to construct the complete binary tree from the set Tp that consists of all termintaing rationals which are appended by a tail consisting of p. Therefore it is impossible to distinguish p from the binary tree. But the binary tree has been errected by using a countable set of paths only. Therefore it is impossible to distinguish, by means of digits, any real number from a countable set of paths.
Conclusion: Cantor's diagonal argument is contradicted.
Comment: This is not surprising, because it is impossible to identify any irrational number by means of its decimal representation, but just this is claimed by the diagonal argument.