Is this mainly a sensor dynamic range difference, or do RAW processors apply brand-specific profiles that affect how shadow colors develop?

Both. The RAW demosaicing algorithm applied may be proprietary (e.g. Nikon NX studio), or it may be third party (e.g. Adobe ACR). Either way, the specific algorithm used can make a significant difference. There was a time when Adobe's demosaicing of Nikon D8xx raw files rendered shadows purple due to a flaw in the algorithm.

But primarily it is sensor dynamic range. At the minimum levels noise becomes prevalent. And there is a minimum signal level the manufacturer determines to be the noise floor of the system. They then code that into the raw file metadata as the black point level (signal level below which should be ignored).

Some programs will apply and hard code the black point in the raw conversion/demosaicing, such as conversion to DNG or proprietary software. Other programs may not hard code the black point and simply set it as the starting exposure level for blacks. I suppose some could ignore it entirely, but I'm not aware of any that do. It is the setting/applying/handling of the noise at minimum exposure that makes most of the differences. Some cameras also apply noise correction/reduction even for raw files; at least at some ISOs. All of this will make a difference in how shadows/blacks are rendered.