Rear hub freewheel and casette
Are you worried that the cogs skip on the hub? I'd consider this extremely unlikely. 

The most likely reason is that the old chain has worn down the teeth of the cassette, especially when you ride on small cogs because the same load is distributed over fewer teeth. If you happen to have the old chain around, measure the length over 10 elements (see my comment above): if 10 elements are longer than 25.9mm (from pin to pin), the old chain damaged the cogs and you need to replace the cassette. The chainring in the front is often still good because it has more teeth and wears down slower.

The image shows the interaction between a new chain and a worn-out cog. It is from technical documentation by Rohloff.

Having watched the video, I do think that the cassette is unusually loose. While it would not cause the chain to skip, it will damage the hub. I found this discussion on this site relevant: make sure to tighten the ring that holds the cassette in place. A mix of road and MTB components could be the root of that problem.