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.
 A: Statistics can be thought of as an application of mathematics towards the rather specific goal of examining numerical data.
In order to understand statistics, you should probably know some mathematics. In a similar manner, in order to understand engineering, you should probably know some mathematics.
But just like in engineering (or any other applied field), there are domain-specific definitions, terms, and conventions that arise extrinsically from mathematics. For example, the $p$-value is a concept that arose to make sense of data. It can be mathematically studied, but it did not necessarily arise as a natural extension of some prior mathematical definition.
A: At my university, the departments are separate. While they have a bit of interaction (i.e. we share a restroom on the second floor), there is very little contact between the two departments. This is true from the earliest freshman classes, clear through to professorial research. The two departments really do not interact. So while the subject matter may be similar ("numbers"), for all intents and purposes, the two fields might as well be totally separate.
One other thing I note is that to get a math degree at my institution, one needs to take a single stats course. In fact, only one stats course can even be counted towards a degree in math. That is the most they will count. But, the stats department requires multiple math courses to earn a degree (BS) in stats. 
My personal view on the matter is that stats is a branch of applied math, much like earning a degree in Korean is a branch of linguistics; but, much of the lingual theory of the latter is avoided in earning the former.  
A: 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.
A: In my university thee are three departaments: Stats, Mathematics, and Mathematical Engineering. The reserch topics of those departaments are different, the hyperspecilization is a general paradigm in the world. 
