is there are way to add the current timestamp/date to the name of a generated target file in the target path rules? I tried with regular expressions like <date> or >day> but this does not work....
No. This requirement is not common and came up here for the first time. It is not possible to define any variable timestamp with the target path rules.
Nevertheless you can achieve what you are looking for with a small macro customization. A system macro that is implementing the callback PassoloApp.BeforeGenerateTarget could loop all target file name properties in the project and change them by adding the time stamp you need.