Stats is not maths? How mainstream is the claim that stats is not maths? And if it's right, how many people don't agree? 
Given that it's all numbers, taught by maths departments and you get maths credits for it, I wonder whether the claim is just half-jokingly meant, like saying it's a minor part of maths, or just applied maths.

How you want to define what is counted as mathematics is up to you.
For me, mathematics is reasoning, and most of the stats classes I've seen have had no reason whatsoever.  It was all memorization and "this is true by definition" when professors didn't know what the background material was.
Statistics can be a great mathematical study, for someone who starts with a strong background in probability and counting.

Aside...
One problem that faces prob/stat is the lack of a good calculus.  It's not accidental that most computer algebras don't have probability support, and when they do it's nearly unusable.  Consider the following word problem:
"You roll a 6 sided dice.  What is the probability that it comes up one?"
Answer: 1/6
"Added information: the evens are twice as heavy (likely to land) as the odds."
Answer: 1/9
An intuitive approach to a calculus of probability breaks the rule:
$$\underline{A \vdash B} \\
A \land X \vdash B$$
Furthermore, in complex problems random variables do a poor job of capturing complex propositional dependencies and independencies.
Simple examples can be explained away with intuition, but the point of a calculus is to avoid the need for intution (as far as verification goes anyway).  I'm sure my opinion is in the minority, but I still think that there is a lot of good progress that can be make in foundations of prob/stat.
