First, it's worth noting that while all the men wear goatees, the styles aren't identical. In particular, Abdullah's spade-shaped dyed beard is very distinctive.
The style is sometimes claimed to be a traditional Bedouin style, but I haven't found any proof of this. Most photographs and illustrations of Bedouins show them sporting either short full beards or just mustaches.
As noted by @user45891, Salafists typically wear long, untidy beards. Indeed, the Washington Post views Abdullah's trimmed beard partly as a political statement:
"In some ways, Abdullah’s beard was the beard of a reformer. According to some interpretations, Islam demands that followers “trim closely the mustache, and let the beard flow.” Well, Abdullah wasn’t having that. His mustache was not trimmed closely — it swam across the top of his mouth like the oil Saudi Arabia so profitably exported under his rule. And his beard didn’t flow, but hugged his chin, falling across his mandible like a waterfall at night. This wasn’t the manic beard of an ayatollah or an Islamic State fighter. This was the beard of a leader turning toward modernity, however reluctantly."
Beards are certainly a sociological signifier in the Arab World. For example Salafists and Muslim Brotherhood members in Egypt are often distinguished by their beards: the former's are long and untidy, the latter's full but groomed.
Finally, the goatee beard style is now also part of the Saudi Royal "brand", which may explain its perseverance. It's worth noting however that the current Crown Prince, Muqrin, sports only a mustache.