To begin, just so we're clear, the use of "missing someone" in this quote does not mean that someone is lost, but rather that you have not seen someone in a while and wish they could be there with you.
That said, let's consider the first sentence if we remove "about."
Missing someone isn't how long it has been since you've talked to them
If we remove "about" in this case, we are left with a statement that, at least in a literal interpretation, simply gives us one of infinitely many things that "missing someone" does not mean. That's not helpful or insightful: it's the logical equivalent of "orange does not mean tree." That's not to say that a native speaker wouldn't be able to make sense of the sentence without "about" in it, but a native speaker would probably infer that "about" is understood to be in the sentence sentence precisely because the sentence doesn't really make sense without it.
Now let's consider the second sentence with "about" removed.
It's that very moment when you find yourself... wishing they were right there by your side
"Missing someone" also is not literally the very moment you find yourself wishing someone were right there by your side. Although wishing someone were right there by your side is essentially what "missing someone" means, the phrase doesn't necessarily suggest a specific moment. Rather, it suggests a feeling of longing.
What the two sentences together are really saying is that "missing someone" really comes down to a feeling you get when you want someone you haven't seen in a while to be with you. But you might get that feeling after you haven't seen your friend for 10 years and then you randomly see a picture of them one day, or you might get that feeling mere minutes after leaving someone that you did not want to leave.
In fact, you could get the feeling before you ever leave someone, if you know ahead of time that you probably will not see that person again for a long time after you leave. In that situation, a common phrase that you might say before you ever left is "I miss you already."