In short: yes, in principle you can absolutely do that.
What I did was essentially very similar: I finished my undergraduate degree where I had focused on political philosophy, then did an internship at McKinsey after I graduated, and did not do another degree afterwards.
In my experience McKinsey Germany also does not have a “target major” as such - while they obviously recruit from business school these people are also usually the keenest. They are just generally looking for really smart and structured students who have business acumen and experience.
Hope this helps :)