Does “evenness” have a deeper mathematical significance than being divisible by 2? I have read and heard many times that “2 is the only even prime number”.
If I was to say “2 is the only prime number divisible by 2” it would be mere tautology; it follows inevitably from the definition of prime numbers, in the same way that 3 is the only prime number divisible by 3, 5 is the only prime number divisible by 5, etc.
Does being even have some deeper mathematical significance than being divisible by 2, or is “2 is the only even prime number” as tautological as “2 is the only prime number divisible by 2”?
 A: In practice, 2 does play a different role than other primes. There are many results (in number theory and algebraic geometry, for example) that are true in positive characteristic different from 2, but fail or are more complicated in characteristic 2. This is because a coefficient of 2 sometimes appears in some formulas (much more often than any other number, except 0 and 1), and in characteristic 2 that becomes zero, but in all other characteristics it isn't zero.
With that said, you will find for example some classification results (projective surfaces for example) that have a general answer, plus one or two more classes that exist only in characteristics 2 and 3, plus a couple more just in characteristic 2. (And, perhaps, also one or two ONLY in characteristic 3.) So 2 is not "really, really exceptional" in that way; 3 is also "exceptional" but less so than 2, and sometimes even 5 is a special case...
A: It is exactly that tautological. By definition, a number is even iff it is divisible by $2$; that's all there is to it.
A: Very relevant is the Strong law of small numbers:

There aren't enough small numbers to meet the many demands made of them.

In the case of $2$:

There are two few even primes.

One could even say that $2$ is prime simply 'because' it is two small to have a non-trivial factor, being just adjacent to one. Similarly $3$ is prime simply 'because' it is right after $2$. Furthermore, every prime is the only prime divisible by it, simply by definition of prime. So $2$ is not special; $3$ likewise is the only prime divisible by $3$.
Also, see the excellent big list of Examples of Apparent Patterns that Eventually Fail and also a specialized question about the specialness of characteristic $2$.
A: "Two is the only even prime number" is only as meaningful as "2 is the only prime number divisible by two".  
But I think there is a deepity to binary states.  If we divide things into "things either are or are not" the pervades mathematics everywhere.  $|P(A)| = 2^{|A|}$ because for each subset of $A$ and each element $a\in A$ either $a$ is in the subset or it is not.  A sequence of "+" and "-" can be represent by $(-1)^{n}$ where the states are determined by whether $n$ is even or odd. Etc.
That's sort of deep.  Kind of maybe.
But I think the deepidacity of it is not that "so and so is even" so much as "2 is the smallest non-unitary natural number".  ... Actually, if I think about it, "2 is the only even prime number" is of absolutely no consequence but "2 is the smallest prime number" is.  
So, I'll go out on a limb and say:  Even numbers are "deep" because they are divisible by the smallest prime number and thus represent of state of dichotomy between "EITHER/OR".
But just how "deep" that is (or maybe it's banally inevetible --- ["but isn't the fact that it is banally inevetible it's self a deep statement about reality?  WOOO! Trippy!"]) is highly subjective.
As a soft question it doesn't get much softer.
