Photoshop lists bits per channel rather than bits per pixel. So, in a 3-channel image (RGB, no alpha) that’s 24bpp, and in a 4-channel image (RGBA) that’s 32bpp.
Aseprite does not have bit depth options, AFAIK. You can save in Indexed Color mode, or in RGBA (32bpp) mode.
Unfortunately I’m not aware of any modern image editors that support lower bit depths, but you might be able to find or write a converter for bitmaps, since that format’s fairly simple and the conversion is easy.