A while ago, I was somewhat bored, so I decided to plot the number of ways that each even could be expressed as the sum of two primes. The evens are on the x-axis, and the number of different ways (where ordering doesn't matter) is on the y-axis.

This graph is strikingly similar to that of the totient function

Is there any good explanation for the similarity of these graphs? Or am I seeing similarities where there are none?

Update

Upon seeing @RobertIsrael's answer, I decided to do some more plotting. This time, it has been colorized so that multiples of 3 are green, multiples of 5 are red, multiples of 7 are blue, multiples of 15 are yellow, multiples of 21 are cyan, and multiples of 35 are magenta, while multiples of 105 are gray, etc, etc.

My conjecture is that the graph here is more similar to $n-phi(n)$, as it seems to be related to the number of divisors $n$ has. This might also explain why the totient graph looks like an upside-down version of the Goldbach graph.

It appears that the numbers divisible by $6$ tend to have more representations than those $equiv 2$ or $4 mod 6$. This is because if
$n = x + y$ with $x <y$ odd and not divisible by $3$, $n equiv 2 mod 6$ requires $x, y equiv 1 mod 6$, $n equiv 4 mod 6$ requires $x, y equiv 5 mod 6$, but $n equiv 0 mod 6$ can
have either $x equiv 1$, $y equiv 5$ or $x equiv 5$, $y equiv 1 mod 6$.

Here's a plot of the number of representations of even numbers up to $10000$, with those $equiv 2mod 6$ in green, $4 mod 6$ in blue, $0mod 6$ in red.