TL DR: Don't discount aftermarket parts as being inferior.
You seem to be mixing up some ideas here. Let me see if I can help clarify:
From my understanding if you have the oe number then you can find the exact part supplied to the manufacturer for your car.
Depending on how old the car is, you can use the OE number to help you find a replacement part for your car. It can also help you find the part from the OEM themselves. As cars age, aftermarket manufacturers will start reproducing parts for vehicles. These vehicles are ones which have usually sold a lot. This is just strict economics. If they can make money selling the parts, they'll recreate (or remanufacture) the parts to sell to the consumer. High wear and regular maintenance parts are some of the first to get reproduced.
However if I search by oe number I can see different suppliers using the same oe number.
In your example, it's doing exactly what you'd want it to do: show you the OE number so you can cross reference it to make sure you're getting the correct part. As @Ben stated in the comments, the manufacturer shows the OE number, but they have their own part number besides. Some parts manufacturers use the OEM number on their items as their numbers. This doesn't mean they were the OEM distributor or manufacturer of the part. I assume they do this to make it even easier for the consumer to find the part.
If one company was supplying this part to the manufacturer
Why do you think anyone is supplying the part to the OEM? For the larger automotive manufacturers, they themselves produce the parts for their own cars. They have the facilities (or the subsidiaries) to produce the parts, which allows them to either sell the car at a lower price or make more profit from the sale of the parts. A lot of manufacturers make parts in plants all over the worlds, then haul the parts to a sub-assembly plant, which puts major assemblies together (like a transmission or engine), then transports those pieces to another facility to make it a whole vehicle.
Or am I understanding the oe number incorrectly, perhaps it means this product was designed to substitute or be a match for oe 44600-S04-A00 ...
While you are spot on in second guessing yourself about the product being designed as a substitute. Using the OE number, as I stated above, is just a way for you, the consumer, to find the part easier. This ensures you have the exact part for your needs.
... therefore it could be of inferior quality.
Just because it isn't the OEM who made the part, doesn't mean the part is going to be of inferior quality. Don't confuse aftermarket as being inferior. This may be true in some cases, but for the vast majority of aftermarket parts out there, they are either of OE quality or better. You can expect these parts to last as long or longer than what you'd get from the dealership.
There's two things you can do to ensure the parts you get should last you like you'd expect. First, take a look at reviews for either the part itself or the company which manufactures them. With the great power of the Interwebz, you can usually suss out who makes crap parts and not buy them. It used to be Chinese parts were hugely inferior, but they've also upped their game and now a lot of the parts which come out of there fit this bill as well. The second thing you can do is once you've found a company which appears to produce superior parts, stick with them. There are some brands which I intrinsically trust. I'll look for their parts in certain categories before I'll use anything else, which includes the OEM.
Is there a way to find out who the original oe supplier was to the manufacturer?
If this is a major manufacturer of automobiles (ie: Honda, GM, Ford, Toyota, etc), they are, for the most part, producing their own parts. They usually don't advertise where/who they are buying them from if they do need to get their products supplied from somewhere other than their own holdings. This doesn't mean they don't buy from somewhere else, it just means finding out who this second party is may not be very easy.
