Is this a pedestrian crossing? It really looks like one, but there are no lights for pedestrians. The traffic certainly seems to think it has right of way. I have a UK driving license, but have never seen or heard of this kind of crossing until coming to London.
Yes, it is a pedestrian crossing. It just happens that this crossing and many others on Fulham Road are not light-controlled for pedestrians.
Item 55 in SI 2016/362, sch 14, pt 2 / Diagram 1055.1 shows the white square marks mark, among many things,
(b) the most suitable place for pedestrians to cross a carriageway within 10 metres of the traffic signals provided for at item 1 [(traffic lights)];
This means the area delimited by the white square marks in the street view is a pedestrian crossing. Having that said, technically, pedestrians can cross anywhere on the road in the UK (with limited exceptions, e.g. motorways, see Rule 7 of the Highway Code / The Green Cross Code). Thus, the question of whether it is a (generic) pedestrian crossing is probably not too interesting.
What does the Highway Code say about pedestrian crossings in light-controlled junctions without lights for pedestrians?
Rule 21 in the Highway Code states the following for pedestrians (all emphasis/omission mine):
At traffic lights. There may be special signals for pedestrians. [...] If no pedestrian signals have been provided, watch carefully and do not cross until the traffic lights are red and the traffic has stopped. Keep looking and check for traffic that may be turning the corner. Remember that traffic lights may let traffic move in some lanes while traffic in other lanes has stopped.
Rule 198 in the Highway Code states the following on using the road:
Give way to anyone still crossing after the signal for vehicles has changed to green. This advice applies to all crossings.
In terms of priority, the latest rule on the hierarchy of road users (Rule H2) states:
Rule for drivers, motorcyclists, horse drawn vehicles, horse riders and cyclists
At a junction you should give way to pedestrians crossing or waiting to cross a road into which or from which you are turning.
You MUST give way to pedestrians on a zebra crossing, and to pedestrians and cyclists on a parallel crossing (see Rule 195).
Pedestrians have priority when on a zebra crossing, on a parallel crossing or at light controlled crossings when they have a green signal.
You should give way to pedestrians waiting to cross a zebra crossing, and to pedestrians and cyclists waiting to cross a parallel crossing. [...]
Note how specific the text goes into (in the 3rd, 4th, and 5th sentence) when describing the types of pedestrian crossing, yet pedestrian crossings without light controls that are not zebra or parallel crossings are omitted. It is only reasonable to infer that, in general, no one gets additional priority around those crossings beyond that covered in Rule 198 and the 2nd sentence of Rule H2.