I got my hands on an IBM PS/1 computer and it provides a 25-pin serial port connector. Whith the help of an adapter I can get a 9-pin serial port on COM1 and transfer data that way. I would like to know wether the PS/1 implements a secondary COM port on the 25-pin connector or if it is left unconnected.
DB25 in its very early stage provided for a secondary port on the same connector (using pins 12,13,14,16,19).
Since then, those secondary and other additional signals became obsolete and even DB25 ports only use up to 9 signals - same as on in DB9 connector.
It is possible that your motherboard provides another port and you would need to provide a cable for it, or you can install a PCI or ISA card with additional ports.
The easiest way to verify would be to check if a system sees any port other than COM0 (or ttyS0), if it does - short pins 14 with 16 and see if you get echo of signals sent to that port.
OR - check in the manual for your exact motherboard - it will give you definite answer.
Two-in-one serial ports on DB25 have been implemented in some unix workstations, but the use of a DB25 connector on 1980s PC style computers for single serial ports was perfectly common and is not an indicator for such a port.
As the above post of the full pinouts shows, the original RS232 standard allows for a lot of optional signals that were not implemented in PC hardware, so the 9-pin connector slowly became the preferred implementation. Serial modems often had the 25 pin connector and came with a 9-to-25 pin cable.
Finding an undocumented (but present from a software/OS view) secondary serial port on some connector on any computer system would be most easily done by sending a constant stream of data out that port (serial ports have no idea whether anything is connected to them!) and checking pins with an oscilloscope. Once the output signal is found, the input pin could be found by connecting that output signal to any pin that is NOT ground, NOT a part of another port, and NOT showing a voltage reading - and checking whether an echo can be established. Using a safety resistor of a few 100 ohms could be prudent here.
Also, the number of implemented serial ports could be estimated by looking for the common line driver/receiver ICs (75189, 75188, 75232, MC1488, MC1489 etc.) and calculating how many signal lines these make possible.