ASME A693 establishes requirements for 316 stainless steel that has been processed following specific methods to achieve a specific form.

S31603 is the UNS number and it has a specific chemistry that goes with it.
316L is just shorthand and not officially the name of anything.
AISI hasn't assigned names for about 50 years.
The ASTM and ASME specifications reference UNS designations, and some of them include a list of common names.
A product specification (SA693) contains product specific information.
How it can be made, how it needs to be tested (destructive and NDT), mechanical properties, and certification and reporting.
In ASTM/ASME there are general specs (A1066 for stainless tubing) that contain all of the tests and then product specs (A249) refer back to the general so that we don't have to repeat everything in every spec.

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P.E. Metallurgy, consulting work welcomed

Just to make sure this is clear: UNS is the designated chemical composition for an alloy, nothing more. These designations evolved from the AISI/SAE designations, at least for steels. Note that UNS designations are now used by the Copper Development Association (instead of the old 3 digit CDA alloy designations) and by the Aluminum Association - I'm sure you will see this standardization extending to other alloy systems as well.

Also note that you will often see alloy grades identified by their UNS designations in many ASTM/ASME (and other) material specifications. You should also note that the specification composition may vary slightly from the generic composition published in the UNS book. When this happens, you are to use the material specification composition to confirm conformance of composition.

In SS and Ni alloys ASTM does not stray from the UNS chemistries.
If you want to put a proprietary alloy into an ASTM you must show that you have applied for a UNS.
30 or 40 years ago the EU agreed to start using UNS for alloys in exchange for cross linking specs.
They haven't started yet.

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P.E. Metallurgy, consulting work welcomed

Thank you all for the replies. If i understand correctly, since the UNS number just specifies the chemical composition, a UNS number can have different product specifications?

Yes, if you get a UNS manual it lists all UNS numbers, any common names, the chemistry, and specifications that reference that UNS.
The specs include ASME, ASTM, SAE (and AMS), AA, ACI, CDA, MIL, AWS, NACE, ISO, and a few others.

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P.E. Metallurgy, consulting work welcomed

Interesting. I'm having a ASME VIII stamped pressure vessel getting constructed for my plant. The info i have from the vendor says it's 304SS. I'm assuming I have to double check to make sure they used the ASME spec for that UNS? I'm assuming they would...

The actual data sheet should call the materials of construction out by SA and then however they are actually identified in those specs.

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P.E. Metallurgy, consulting work welcomed