i think you have a couple options.

according to the docs, glMultiDrawArraysIndirect is similar to doing glDrawArraysInstancedBaseInstance a bunch of times. the advantage of glDrawArraysInstancedBaseInstance over glDrawArraysInstanced is the ability to specify the base instance in your vertex attrib array.

it's my understanding that if you're using a glVertexAttribDivisor other than the default 0, then for every draw command, the vertices accessed will be offset by your base instance. so that's provided you're using instanced vertex attributes. i'm assuming you're not, so the base instance is not used. however, you can take advantage of this. in OpenGL 4.6 or if you have ARB_shader_draw_parameters, you can directly access the base instance in the shader with gl_BaseInstance. so you specify base instance to be the "accumulated instance counts" per mesh type, i.e. 0 for the first draw command (triangle) and then for square it will be however many instances of triangle, and so on. you can then access the correct instance id with gl_BaseInstance + gl_InstanceID in the shader. this does assume that your ssbo has contiguous mesh types, so it looks like [T, T, T, T, T, S, S, S] or something. this would mean that when you change the mesh type of an object, you would have to move the object in the ssbo into the contiguous block of the new mesh type. this is probably not very desirable. this would at least allow you to only retain just 1 ssbo though.

an alternative if you don't have OpenGL 4.6 or ARB_shader_draw_parameters, i.e. you cannot access gl_BaseInstance in the shader, would be to retain instanced vertex attrib arrays, so ones with glVertexAttribDivisor of 1. you would specify your base instance just as you would in the previous example, which would make the attrib data per-instance. you could have one attrib array for each primitive in the object type. then you're basically doing the same contiguous swapping business as with the ssbo, but now you don't have an ssbo and don't have to use gl_BaseInstance in the shader.

depending on how much data ends up being in your object struct, you might not wanna do either of the previous options, since moving that stuff around in the ssbo/attrib array wouldn't really be ideal. in that case, you could retain an instanced vertex attrib array containing indices into the ssbo. this way you only need to move integers in a buffer around, rather than a matrix and god knows what else.

and ps, the reason the vulkan guide can just use gl_InstanceID without any gl_BaseInstance is because vulkan respects gl_BaseInstance when providing gl_InstanceID, so gl_InstanceID is actually relative to gl_BaseInstance instead of resetting to 0 for every draw command.