Earlier quoted context omitted.
The wording in the article is specifically: > requires only 20 watts of power per liter. Whether the article is accurate or not I think you missed that.
The point is that 7W/hour is nonsensical. It's a non-unit and physically doesn't make sense in any context. The power is independent from the volume. Total energy use is volume dependent. 20 Watts per litre is mix of units that's nonsensical. Watt is Joules (Energy) per second (time), i.e. power. Either time is irrelevant, that is the device draws 20 Watts of power, in which case you can ditch the litre. Or the power…
If this is correct, they're saying it takes 20 watts (meaning Wh) to deliver a Litre of water, over three and a bit hours. The time is provided just obfuscated by their "we deliver 0.3L/hour" statement, so by saying 20W/L it does sort of make sense within the context.
~6W/hour (if it means we use a total of 6Wh) would be different than claiming the system runs off 0.1 Watts. Given the size of that solar panel, I suspect they're claiming they are using ~6 Watts for a period of one hour to produce 0.3L. Alternately, they're just flat out wrong and they're using 20 watts for one hour to produce 0.3L.
It could be worded better but I think this is what they're trying to say is that they're consuming 6W.