Are there proprietary differences that would cause one brand's crown-race to interface improperly with the bearing in another manufacturer's headset, assuming all parts were made for the same diameter steerer and the same steerer type (straight/tapered)?
In the history of bicycles, no they are not broadly interchangeable in any way. Most bikes in the world or in history have a headset where the crown race is designed to mate dimensionally with the cup in a specific way for sealing, to put the ball track in the intended place on both the cup and crown race, etc.
Among basic looseball/retainer headsets, there are a few very common retainer designs for 5/32" balls. A cup/race mismatch that uses the same retainer will often be able to physically work, but the sealing won't be right, and it is possible that there could be physical interference as well depending on the design. It is best avoided given the choice, but that said there are many bikes in the world rolling around more or less fine with a mismatch of this type.
In the current era there is broad interchangeability among IS headset crown races of a given lower steerer diameter and bearing crown race contact angle, to the point where there are almost never issues mixing and matching. Note how integrated crown races have become common on aftermarket forks. That wouldn't be possible if the standard was designed for anything to really matter but the angle and the size. Conversely, the entire IS standard has the massive drawback that it was designed to not bother having very effective sealing, or the option to have it.
IS42 and IS52 can both have 36 or 45 degree crown races. The frames can take bearings that have either. 45 has generally become more common, and that's the size that most integrated crown race forks use. But, because all 4 permutations exist, there are times you have to either change the bearing or the crown race to make things work depending on the situation at hand.
Yes, there are differences that can prevent this.
Headset bearings come in several angles on the inner and outer diameter of the part. 36 degrees and 45 degrees are common. The angle on the crown race must match the angle of the bearing chamfer, or the stack will not fit together.
For example, Cane Creek typically uses 36 degrees on the outer diameter and 45 degrees on the inner diameter. Campagnolo typically uses 45 degrees on both. To guarantee the headset fits together, use the same brand crown race as the rest of the headset.
The other two answers are fine but they don't give a fair picture.
You have effectively asked two questions:
Are crown races interchangeable? and Are there differences between specific headsets?
So the answers in brief are "Some are" and "yes."
Depending on the type of headset, the crown race is often a standard part between manufacturers, so far as the bearings are concerned. There are a large number of 1" or 1 1/8" lower cups that are designed around the same caged bearings that are commonly available as replacement parts. With these, the differences are only likely to be in the sealing system, if present, and even then, may work without a problem. Various bearings (eg NJS/Track) are designed without a seal as dirt ingress is not considered very likely or a problem. It is easy and cheap to check with these parts that they will work together.
Problems arise with types that use unusually sized or over size bearings (campagnolo etc, dia compe g-cup etc), needle bearings or other specialist types. These need a complete setup.
Cheaper types are generally all the same design. and you can play around with individual components to an extent.
The more modern cartridge types are absolutely interchangeable providing that you don't mind potentially losing an extra seal that not every headset maker includes and that you select a bearing that matches the angle of your crown race. Bearings exist for the common angles and can be different top and bottom, so the part you need is generally available.
It helps to understand how bearings are measured and classified in order to mess around with these systems but there is alot of room for getting a working combination.
OBVIOUSLY You should be confident setting up a headset before using a "custom" combination, so you can tell if it feels "wrong" or right, safe or unsafe. Don't ride an unsafe headset, hey! You'll get gorse in your hair, at best.