Punjab is Persian: panj ab, five waters. The name is Mughal-period and the region had other names before it.
Sapta Sindhu in the Rig Veda, the seven rivers. Pentapotamia to the Greeks, five rivers, the same idea in another language. The Sanskrit Panchanada. Under the Delhi Sultanate it was simply Lahore province.
The five are the Jhelum, Chenab, Ravi, Beas and Sutlej. The Indus is the sixth and is not counted, because the five are the ones between which the doabs lie.
Since 1947 there has been no Punjab with five rivers in it. Pakistani Punjab has the Jhelum, Chenab and Ravi and part of the Sutlej. Indian Punjab has the Sutlej, Beas and Ravi as a border. The name describes a region that no political unit now corresponds to.
Indian Punjab after 1966 is about one-eighth of the area of the 1901 province. It is still called Punjab.
The word is used by everyone in it, in both countries, and it means something larger than either state that carries the name.
Panj ab is five waters. Before it the region was Sapta Sindhu, the seven rivers, in the Vedic hymns, and the Greeks called it the Pentapotamia.
The Persian name entered use under the Mughals as the name of a subah and it has been the region's name in every administration since.
What it names has changed each time. The Mughal subah, the Sikh state, the British province, West Punjab, East Punjab and the state after 1966 are six different territories with one name.