I've never been a fan of the phrase "strong not skinny," and for a long time I couldn't really articulate exactly why, but I think I got it now. When folks say strong not skinny, I believe they are trying to move away from scale-based outcomes towards training for overall wellbeing and, well, strength. That is a principle that, broadly, I can get behind.
The problem with "strong not skinny" is that it often comes to mean striving for increased muscle mass, decreased body fat, and achieving an overall "lean" appearance, which may not necessarily look "skinny." It just creates a new and different body hierarchy, reinforcing the idea that SOME bodies are good, and deserving of love, dignity, and respect, and OTHER bodies, well... They're just other.
If all we're doing is reinforcing the idea that one of these bodies is inherently better, healthier, more desirable, worthier, even stronger than the other, then for me, that ain't it.
I'm going to push it a little bit further though... Why (and this may sound strange coming from me, a powerlifter and personal trainer) should strong always be the goal? Don't get it confused, I love strength, and I love helping other people get strong. That said, strength can look, feel, and mean a lot of different things. I love working up to one rep maxes with a barbell. I have clients who want to be more mobile and be able to carry their essentials without fatigue, but don't want to touch a barbell. That is what strength means for them. Sometimes, we'll do training sessions or classes where we're doing a shit ton of reps with low weight. That is a different kind of strength than one rep max strength. None of those are inherently more valid, worthy, or important than any of the others. For people with disabilities or health conditions, strong may mean something quite different.
We talk about health not being a moral obligation. Strength is also not a moral obligation. You're not under any obligation to try to get strong. Will I talk to you about the psychological, physical, and overall health benefits of getting stronger? Hell yeah, I will! But you don't have to do it. I believe in body autonomy, so YOU get to choose how you want to live in your body. After all, it is your body!
Sometimes these catch phrases pop up intended to empower folks, but there are so often layers to things that can't be captured in a hashtag.