No, that has nothing to do with IO::Socket::INET. It has to do with <$sock> also known as $sock->readline(); which won't return until either $/ is read or end-of-file. Killing the client closes the socket which sends end-of-file (as would using shutdown).

This is part of the reason one should normally not use <$sock> but instead should use sysread (or similar). Even if your protocol adds newlines on the end of each "packet", using <$sock> makes it hard to debug what broke when things go wrong.