I think I see what you're saying - however I'm not hearing anything indicating a different aerospace-specific
material, but rather aerospace-specific
process. The raw stock with certs has a different P/N than that without certs, but not it's not a different material. For instance, we might procure aluminum 7075, which has a published spec in the form of ASTM B209 (and several others, this is one I've seen called out in drawings commonly). 7075 is available in multiple different tempers - you can get 7075-O (not heat-treated), 7075-T6, 7075-T651, and a few other less commonly used ones. When used for aerospace, that material will generally come with a cert from an independent test lab showing that a specimen from the batch meets the yield strength, ultimate tensile strength, yield at rupture, composition, and other critical properties. At the end of the day, the piece of round bar or sheet is the same thing you'd purchase otherwise, but you've paid quite a bit extra to be SURE that it's exactly what you expect. The same applies to steel, titanium, nickel alloys like Inconel or Monel, tungsten, magnesium, and pretty much everything else I can think of.
Following procurement, we might do in-house testing before machining (composition with XRF, physical properties with a tensile tester), possibly our own heat treatment (e.g. 13-8PH, 15-5PH, 17-4PH are "precipitation hardening" steels, generally delivered soft), and surface treatment (passivation, conversion coating) before delivering a finished product. None of this is unique to aerospace either, although it's certainly unlikely you'd want to spend the money for it otherwise.
So yes, you procure a different item, possibly from a different supplier, but physically there's unlikely to be any difference in my experience. The exception I'm aware of would be electronic components, particularly semiconductors, which are manufactured using different processes for radiation hardening (e.g. sapphire substrate). Export control like ITAR/EAR aren't really about aerospace but rather restrictions imposed by the US Government.