Does the AMD 5600G support ECC memory?

No. AMD locks ECC support behind the PRO SKUs on the 5000G series. Some motherboard BIOSes might give you the option, but it’s not validated and may not actually correct errors — it could just be reporting them. If you want real ECC for your NAS, buy the 5600G PRO or switch to an Intel platform with unbuffered ECC on consumer CPUs.