I would say Adobe After Effects is the software to do these kinds of animations. It's definitely not easy though and takes a very solid foundation in motion graphics. I've done work like this for Coca-Cola, Voya, Delta Airlines, among others, and it took 3-5 days, depending on exactly what the client was wanting.
No question you can do this in AE and most people probably do. I have done stuff like this in Apple Motion, which I find more intuitive and simpler to use than AE, and what can help someone who is inexperienced at this is, in either platform, MotionVFX.com offers a number of very handy plug-ins and pre-built templates that make the cutesy, bouncy, squeezy, stacking effects all pre-done for you, pretty much drag and drop. The labels and graphics too. They also offer clip media element packages, people, cars, houses, etc. so your first step of building up all those items and characters and "props" in Photoshop is eliminated or made much less tedious.