This is highly likely to be a dumb question, but can each dish only communicate with one object at a time?

These are highly directional antennas that are pointed at the vehicles they’re communicating with.

Yes, though sometimes we use something called "bent pipe" where we communicate with something like MRO who is then communicating to an asset on the ground at Mars, so technically we're talking to two assets, but not "really"

The DSN stations can receive from multiple spacecraft at the same time, but can only transmit to one. If you look at the DSN status station and click on the details sometimes you will see them in "MSPA (Multiple Spacecraft Per Antenna) mode". Looks like they can do 4 at the same time [0]. You will typically see this for Mars spacecraft where there are multiple within the beamwidth of the antenna.

It also looks like they have proposed some options to provide additional downlink by doing raw recording of the intermediate frequency with a high bandwidth receiver and doing the demodulation with additional processing to get telemetry/data for more spacecraft [1]. So they could support getting more data, but not necessarily "realtime" data.

[0] https://deepspace.jpl.nasa.gov/dsndocs/810-005/206/206D.pdf

[1] https://ipnpr.jpl.nasa.gov/progress_report/42-200/200B.pdf