Can I retrofit a ground wire to my oven circuit?
You can retrofit ground. See many answers about that - e.g., What does this answer about old residential home grounding mean?

Options include running a new separate ground wire or connecting with another ground wire. However, while you may be able to use a smaller than 6 AWG wire for the ground (10 AWG according to Harper's comment), you can't just connect to a 12 AWG or 14 AWG ground from a 20A or 15A circuit - that's not quite big enough.

As far as all the wires being together - ground is the exception. That's because ground only carries current in an emergency (and then hopefully only long enough for the overcurrent situation to trip the breaker). However, one key to that is make absolutely certain that any neutral-ground bond in the oven is removed. Sometimes a neutral-ground bond is included in the oven but that is only used when connecting without a separate ground to the panel (which should not be the case on any new installation).

As far as why this is all even an issue: When grounding was added as a requirement for most residential circuits, appliance manufacturers (specifically electric ovens/cooktops/ranges and clothes dryers) were concerned that requiring a potentially significant amount of electrical work (e.g., if the wires were not in conduit and the path not easily accessible to run a ground wire or a new cable) would affect sales of new appliances - you would replace it if it was broken and you had no way to fix it, but you wouldn't "upgrade" and you would fix rather than replace when possible. In other words, money. So they got an exception to allow ground to share the neutral connection. In certain situations this provides the same safety benefits of a separate ground wire. But in many other cases it does not. At this point (and for many years now), new installations and changes to existing installation require full separate grounding. But the manufacturers are still resistant to losing any sales due to lack of a ground wire so they continue (legally) to sell appliances that can be connected either way, and in many areas it is totally legal to keep using a ground-via-neutral provided it is a simple appliance replacement, but not recommended.

As far as neutral-ground bond in the main panel, the basic idea is that neutral &amp; ground should be connected in exactly one place. In fact, they need to be connected for certain aspects of the entire electrical system to work properly. But having them connected in multiple places ends up defeating the purpose (partially) of ground wires.