We need to disambiguate some terminology first. It's obvious what a chainring is. For a long time, I think that when people said "crankarms," they meant the crank arms plus the spider - that is, the arms that you bolt the chainrings to. The arms and spiders were forged (aluminum) or laid up (carbon) together. The picture below is a Shimano Dura Ace 9000 crankset.
Direct mount chainrings are different. Here, the spider is forged or stamped together with the chainring. "Crankarms" in this context refers to the arms without the spider. The picture below is a Shimano XTR M9100 chainring. It fits via a splined interface to the crankarm with a lockring.
I'm not sure that all BMX chainrings are direct mount; a quick Google search shows some that would bolt to a traditional crankset (with spider). I know that direct mount is much more prevalent on MTB cranksets than on the road. SRAM has shifted to direct mount cranksets starting, I think, with their current generation of road cranksets (Red and Force 12s AXS). Shimano and Campagnolo still use the traditional arrangement.
So, broadly speaking, there are two types of cranks, the more traditional type and direct mount. For traditional cranks, the standards in question would be the various bolt circle diameters; current generation road have sort of settled on 110mm, but Shimano and Campagnolo have different 4-arm geometries. SRAM used 5-arm 110mm cranks for some time before changing to direct mount.
For direct mount cranks, the standards in question are the splined interface. I know that SRAM has one standard. I believe that Race Face and Easton, which have the same corporate parent, share a different standard. I can't remember if Cannondale either have a proprietary spline standard, or that Race Face/Easton share it (in some sense, we can also thank Cannondale for press fit bottom brackets, as they were an early mover). I am fairly sure that Rotor, which is a niche crankset and chainring maker and a proponent of oval rings, has at least one of its own standards. In their case, that standard enables you to rotate Rotor's own oval rings to your preference. Shimano also has its own spline standard.