The short answer is: most well-built urban benches can handle kids jumping on them, but they are not designed as playground equipment. I have watched plenty of kids climb, stand, and bounce on city benches without the bench collapsing. That does not mean the bench is indestructible. "Strictly for sitting" is more about preventing wear and accidents than about the bench breaking the moment a child jumps.
A typical commercial outdoor bench uses a steel, cast iron, or thick aluminum frame with wood, metal, or recycled plastic slats. These are built for public spaces, so they are much stronger than indoor furniture. Many public benches are rated for static loads between 500 and 750 pounds. Static load means weight that is not moving. A jumping child creates dynamic load, so the peak force can be two or three times their body weight. A healthy metal bench that is properly anchored will usually absorb that, especially if the child is small or only jumps for a minute.
What actually fails first is rarely the frame. The fasteners, slats, and surface coating usually go first. Repeated jumping can loosen bolts, crack wooden slats, chip powder coating, and invite rust. If a bench is already rusted, wobbling, or sitting on an uneven base, the risk goes up.
There is no universal rule that says urban benches are strictly sit-only. But a bench is not a trampoline. If your child jumps on it occasionally, the bench will probably survive. If it becomes a daily habit, expect accelerated wear. Check for a heavy frame, thick slats, tight bolts, and solid ground contact. Avoid jumping when the bench is wet or icy, and watch the edges because they can be hard on little knees and heads.
So the bench will likely hold up, but the better advice is to use it for sitting, standing, or supervised climbing. That way it stays safe, functional, and lasts much longer.