I would like to connect one USB Headset to two separate computers, hearing the audio of both at the same time. USB Switches exist, however they only allow one computer to control the headset at a time. It most certainly needs to be a special device that knows how to handle sound data and mixes it from two different sources. Does something like this already exist? If possible, buying one is the preferred option however I may also invest some time and tinker it with a raspberry pi if some tutorial for this exists.
You want a "Sound Mixer". If you are working with analog audio signals, there are several options: Here is a cheap one that has 4 inputs and here is a battery powered option with better quality that has 2 inputs.
Unfortunately there are not any devices that share a USB connection between multiple hosts. That is just a limitation of USB. USB was designed as a master-slave system for peripherals, as such, it has a strict "Tree Network" topology. This means, no matter how you slice it, multiple hosts (computers) can't share a USB device. Similar issue on this recent question.
Because it is common to want this functionality, there exits several software solutions. I know this is outside the context of both this site and what you want, but its really the only option. This is where the Raspberry Pi you mentioned may come in.
You can find many options by searching for a way of sharing either the USB device, or the sound source over a network. A good example of this is the program Synergy. The original version of the program (there is a second more complicated version now) shares your mouse and keyboard between multiple computers by connecting your peripherals to one computer that acts as a server and sends its kb/m inputs over to another computer through a network. The program running on the second computer decodes this information and presents it to the operating system as a kb/m. The second computer can't access the keyboard or mouse directly, it only gets the keystrokes that are sent to it.
To do this with Audio there are programs like PulseAudio and Virtual Audio Cable. I can't say I have experience with either program so you may want to do your own research on them. You could use a Raspberry Pi to act as the host that you plug your headphones into, but you will probably not see any benefit over letting one of your computers be the host unless neither of those computers will always be available.