Baking is for extracting rendered data into uv images. In the history of blender this wasn’t always posssible because of the workflow and/or initial architecture of the renderer (that’s the reason why even today it’s… not easy) in eevee it isn’t even possible yet (i almost always have to look up the docu). So baking has to be handled extraordinary special… because the results aren’t always what someone wants and have to be checked… of course that’s the reason why there are so much bake addons… on blender market and elsewhere, even free ones hosted on github. The fact that standard baking isn’t easy reflects also in the high number of questions here on BA… lately someone had difficulties with some texture bleeding between arm and body which made me curious because those were two different materials and i wondered why they weren’t baked individually (or i don’t know yet to bake them in one step)? You also seem to struggle with baking having multiple baked maps to save and seek for a posibility to save them fast. And as i proposed: isn’t it possible to settup a texture map in ever to-be-baked material so bake them all and save them all ?
And after all i didn’t said: “just save it” i asked why saving after baking is so tedious (?) for you. On the other habd i assumed you searched allready for some solutions/answers and if this is so precious for you why not pay 15 bucks ?? So here https://inspirationtuts.com/material-baking-addons-for-blender-must-have/ seems to be 3 addon which batch baking one is free on github… (There is even an (simple?) attempt to bake textures using the blende API writing a C++ program, but it seem to be that the whole scene is build via programming…)
Nevertheless i also learnt a bit more about baking, so thank you for this fruitful conversation.
Edit: (hmmm three followup aren’t allowed) i just tested Principled Baker… literally just hit bake See Principled Baker - Bake PBR textures with a few clicks - #232 by Okidoki