Can a Mac mini connect to a 5.1 speaker system?

This answer addresses the OP's original question. This was before the question was modified by other users. The original question is given below.

Can a Mac mini connect to a 5.1 sound system?

Is there any gadget or device that can plug into a Mac Mini's headphone jack that can then connect to a 5.1 speaker system?

Starting with the 2006 Intel Mac mini models, optical digital audio output could be accessed from a mini‑jack. This mini‑jack could also provide analog audio output for headphones.

An example of a 2 meter "Toslink to Mini Toslink Digital Optical SPDIF Audio Cable" is given here. The Mini Toslink end of the cable would plug into the 3.5 mm mini‑jack on the Mac mini.

The end shown below plugs into the Mac.

Below is an image of the other end of the optical cable.

Starting with the 2010 Intel Mac mini models, a HDMI port was added that supports multichannel audio output. Support for optical digital audio output from the mini‑jack continued.

Starting with the 2018 Intel Mac mini models, support for an optical digital audio output from the mini‑jack was dropped. Support for Multichannel audio output from the HDMI port continued.

All the Apple Silicon Mac mini models do not support optical digital audio output from the headphone mini‑jack. The HDMI port supports multichannel audio output.

The table below summarizes what was stated above. Note: Some years Apple did not introduce a new Mac mini model.

To convert the digital HDMI audio output back to digital optical audio output you would need an external "gadget". For example, there is the Insignia™ - HDMI Audio Extractor with 4K @ 60Hz / HDR Support. The front is shown below.

The back is shown below.

There are also USB devices with digital optical jacks. However, such devices can often have worse issues syncing the audio with HDMI video than devices where the HDMI passes through.

HDMI is standard for multi-channel audio on Mac.

HDMI port supports multichannel audio output
Mac mini - Technical Specifications

You may have seen systems that have a bunch of 3.5mm jacks that are meant to connect to a desktop style surround system. If you need this style of interface, you might want a USB sound card (sometimes called a DAC) with 3.5mm surround output like this device from Cubilux or the Creative Sound Blaster X4.

If you need to separate the audio signal out from the HDMI port, another option could be an HDMI audio extractor. These devices sit inline with your HDMI signal and split the audio out into a new optical audio (TOSLINK) port, which will carry surround sound. See this device from OREI.

TOSLINK will only carry classic, compressed Dolby and DTS audio, although macOS does not seem to support uncompressed audio passthrough yet.

Unlike an HDMI connector cable, a TOSLINK optical fiber connector does not possess the bandwidth capacity to carry the uncompressed audio signals of Dolby TrueHD and of DTS-HD Master Audio
TOSLINK - Wikipedia

Do ensure that it will work with your system. I have not used these specific devices, although I have used other USB and HDMI audio interfaces in the past.

If your configuration relies on directly connecting the built-in HDMI port to your display, this may mean getting a USB-C AV adapter that adds an additional HDMI port for your audio device. I have been able to send only audio over HDMI from an Apple TV set-top box to an AV receiver with no TV or display connected.

It may also be worth enabling HDMI audio passthrough in macOS applications like Apple TV, as this will send the media's encoded audio signal directly to your surround system for improved quality and compatibility, at the cost of being able to mux other system audio into the primary audio device when media is open.

The original poster asked about using the 3.5mm headphone jack for surround sound, although this answer has been revised to better address the current question. The TRS connector used for a headphone jack simply does not have enough contacts to send that many audio channels without some kind of non-standard (and lossy) signal format. This is why a headphone jack that supports a microphone has an additional ring on the connector, a TRRS connector.