The rust can not be originating from a brass tee. Rust is "Iron Oxide" and only occurs on metals or alloys that contain iron.

I've removed the T and ran my finger inside it and there's rust. The T is brass colored, not copper. I'm assuming it's junk pot metal from China that is composed of God knows what. Is this common or possible?

Regardless of where that tee originated it must pass minimal standards as set forth by the ASTM & ANSI.

All fittings listed as brass fittings must be solid brass and brass is an allow of copper & zinc .

On the other hand, even if it were potmetal as you would like to convince yourself, Potmetal does not contain any iron, therefore the rust could still not originate from that fitting.

It is far more likely that the rust is resulting from iron in your water. In regions where there is a high content of iron in the water it is very common to find a rust film on the interior of water heaters because the heat acts as a catalyst separating the iron from the water.

Make sure a magnet does not stick to the T.

That will at least mean it is not plated steel (very unlikely though).

It does sound like water or a defective tank lining.
I have seen a very few over the years that had large areas that did not have glass lining.

Thanks. I'll try the magnet test. The tank is glass lined. Ariston GL-4.

I looked at our water report and can't see iron content as an element they test for. I've been told we have hard water... its 250ppm. Is that related?If it is truly iron separating out from the water because of heat, why doesn't it happen when the tankless water heater, the builder installed, heats the water for the rest of the house?