In order to run multiple instance of the Appium server, once might assume from the documentation that setting a unique port number is sufficient. It turns out this is not the case.
The reason is that if no TMP directory is explicitly set (on the Mac), then /tmp is used. This causes problems since the instruments trace file, the run results, and the instruments_sockets file are being shared between instances and this causes unexpected results and lost log data. Manually setting the TMP dir with each instance alleviates the problems.
A random name should be used for each new server instance.