That looks like a glass mug. And a mug can very easily be interchangeable with the term 'cup' in normal speech. Regarding the image posted, it might commonly be referred to as a 'mug', 'cup', or even a 'glass' (assuming it's made of glass).

Technically it is a mug. Cup doesn’t Reallt have that handle. But there are times I have called that a cup also.

It is a mug. A cup would usually have a matching saucer. In recipes you will see 'add one cup of water', that is a measurement equal to 250ml.