I finally found the root cause for ElixirLS not working for the existing project.

$ MIX_ENV=test iex -S mix
Erlang/OTP 27 [erts-15.2.3] [source] [64-bit] [smp:16:16] [ds:16:16:10] [async-threads:1] [jit:ns]

** (File.Error) could not read file "/home/jan/Workspace/project-b/config/test.exs": no such file or directory

I renamed test for that project, but ElixirLS requires it. Adding config/test.exs solved the issue for me.