I really wish more systems would adopt the "copyindent" pattern, where indentation is simply copied from the existing line. I've primarily developed in vim for 15 years and have similarly minimal `vimrc`s (no plugins), but some of the first configurations are always:

  set autoindent
  set copyindent
  set preserveindent

This should be compatible with basically any language (other than some esolangs like "whitespace") and any tab/spaces scheme. There's no need to have per-file/project configuration, as long as you're okay with pressing space 4 times instead of tab in the projects that use 4-space indentation.

I'm not ok with that. I use EditorConfig so as not to have to remember which project uses tabs and which uses spaces.

The ftplugin for the make filetype includes setlocal noexpandtab, which should take precedence over what's in your vimrc.

  % vim Makefile
  :verbose set expandtab?
  noexpandtab
    Last set from /usr/share/vim/vim90/ftplugin/make.vim line 15
  
  % vim file.c
  :verbose set expandtab?
  noexpandtab
    Last set from ~/.vim/vimrc line 56

but I think that may not work as intended for OP as they don't have "filetype on plugin".

It’s been a while but I seem to remember being able to configure different tab behaviour specifically for make files.

It surely can. If I have to casually mess with a Makefile, I'll just remember to use control-v TAB as needed, or vi yank/put another line with a proper tab and edit. In the very rare case I'm doing more serious work, I'll temporarily turn it off.

You can change it per filetype. For example, I have a similarly minimal vimrc that defaults to sw=4, ts=4 and expandtab, but with the following additions:

  autocmd FileType go,make,sh set noexpandtab
  autocmd FileType js,json set sw=2 ts=2

Edit: I think you might also need the following to make it trigger the FileType event when you open a file in the first place:

  filetype plugin indent on