A host that accepts connections from a client through the serial port is no different than a host that accepts connections from a client over TCP - in that it may be possible for a malicious client to exploit a vulnerability in the host and compromise the host.
For example, a host that accepts incoming connections through the serial port must be protected against buffer overflow attacks (which were very common in servers that accept incoming TCP connections, and still are to some degree), or else a malicious client connecting to the host through the serial port could potentially cause a buffer overflow. As another example - if there is a database involved in the service that the host is running, then sql injection attacks (which are notorious for being deadly when it comes to data exfiltration) may be possible, regardless of whether the connection is by TCP or serial.
Having said that, an attack through the serial port may be less likely in the real world, simply because of proximity reasons. Whereas an attacker launching an attack through TCP could be halfway around the world, an attacker launching an attack through a serial cable would be at most 50 feet away.