Typically in the EU (or at least here in The Netherlands) you can be fired on the spot (without a notice ahead of time) in the first month (usually referred to as a "trial month"). I suspect this is what SoundCloud took advantage of.
OP should ask a lawyer. I think this is not legal.
What is the process for quitting like?
If the employee and company part on bad terms, they can still part pretty fast. There's no mechanism to force someone to come into work (besides the threat of a bad reference) and if they did come in, there's not much stopping them spending all day distracting people and moaning about how much they hate the place.
If the employee and company are parting on good terms, the employee will hand in a written notice then work their notice period, with the possibility of ending it sooner by mutual consent. The employee generally wants to finish up what they can and hand over what they can't because they like their job and co-workers.
Needless to say, if you're a senior guy who's been at the company for years, there'll be more stuff you know and responsibilities you've assumed that you need to pass on to other people. For junior employees and recent hires, there will obviously be much less.
Real question is it worth chasing them for it.
Two weeks notice during probation period. - What's more relevant:
If they actually signed a legal contract they also need to provide him with a written and signed contract termination. As far as I understand a signed contract exists and no written notice has been delivered. The notice period is counted starting from the day of the delivery of that physical mail.
https://www.anwalt.de/rechtstipps/die-kuendigung-vor-arbeits...
2) After they've quit, moved, uprooted their life, but before they actually start working -- fire them.
3) Win!