Interesting question. Yes I do think there are "personalities" to schools, driven largely by the principal. My three sons have all attended the same early elementary school (grades K-2) and have had three different principals. When my oldest son was there, the principal was nice enough but retiring. She was very clearly short-timing her last year or two there. School was fine, but nothing spectacular. The next principal was a nice but rather cold and rigid woman. Her nickname was "Lilith" (as in Kelsey Grammar's character's wife from "Cheers"). She was a good administrator but no fun. The answer to everything was no. Very overly concerned with things like what the insurance policy would frown upon.
The newest principal is starting his 3rd year there and he's simply awesome. Young, energetic, has children the same age as the students, and thinks that kids learn best when school is FUN! The answer to everything is "sure, let's try it!" In the first month of the school year, the kids all create and ratify a constitution that becomes the guiding document for behavior for the year (of course it basically says the same thing each year, but the kids have buy in and own it, and hold themselves and each other accountable to it). He let parents transform an empty side yard into an amazing vegetable and flower garden that is an outdoor classroom. The cafeteria serves food from the garden and partners with local farms for healthy foods and the kids actually eat it. The kids can bring their snow gear in and play in the snow during outdoor recess in the winter. We have a welcome back picnic before the start of the school year and lots of great family activities throughout the year that are really well attended. Every morning, the announcements start off with music, birthdays, and an inspirational quotation read by a student. This school is the most enthusiastic place I've ever been, and it starts at the top. Energetic, passionate principal = energetic, enthusiastic, joyful teachers = kids who come to school ready to learn because they love it there = parents who will do anything we're asked to do to make this work.
I think that in the older grades, the principal and assistant principal have a huge influence on how things run and what is and isn't tolerated in school. When my oldest kids were in middle school, they both were involved in issues that came up with bullying, rumors and other behavioral issues. My kids were either victims or witnesses. The assistant principal took things very seriously and really backed up the talk about what wasn't tolerated with action. One day my step-daughter was having an issue and was looking for the VP, who wasn't available, so the principal invited her to come to his office and talk. She told him what was going on, he gave her some good advice, kept his word that he would keep and eye on the students involved and then the problem went away. So there were issues, but they were addressed.
I think that the observation about grades is true too. The incoming 8th graders ("the millenials" - born in the year 2000) have had a reputation for trouble since they were in Kindergarten. It's a huge class - 30% larger than a typical class in my town - and that class has had issues come up a year or two before they normally do. So rumors of sexual stuff in 5th/6th grade, of drinking in 7th grade, etc. Certainly it's not the majority of kids, but there are quite a few in that grade that have trouble written all over them.