Can a production Oracle DB server also have a dataguard active standby for a different production database? If you have a production oracle database (PRIM1) protected using Oracle Dataguard to a standby (STBY1) physical server in a different data centre, the standby server needs to be a similar specification to the primary server, so that in the event of a disaster it can become the primary.

This setup means that you have redundant hardware 'just in case'.

If you have another production database (PRIM2), could you install this on the same physical server as STBY1 of the first application and install the standby for this application (STBY2) on the primary server of the first application (PRIM1).

This setup would mean maximising use of the hardware and in the event of a failover, both primary servers might be running on the same machine so you might need additional memory and/or CPU available to provision in the event of a disaster but you would be using the hardware most of the time. 

Any thoughts on the issues a setup like this would create ?
Yes, no issue at all. This is done over and over. For instance two servers taking each others database(s) in standby mode.