I'm getting this from a Spanish SEAT enthusiast website, so bear with me.
http://seat600.mforos.com/519035/4236018-numeros-de-bastidor-y-motor-lista/
It seems that 600 production began in Spain in 1957 with the A series 600.
From 1960 the 2nd series changed to B with a few variants - BA Saloon, BB Truck, BH Convertible BU Bare chassis for coachbuilders etc.
The chassis VINs ran sequentially from the first A series and the chassis types were all intermingled so it was possible to have, for example, BA-000.100 then BH-000.101 then BB-000.102.
All of the series produced after the first bear the leading B letter. To complicate matters, it seems that the last series (1968 to 1972) where the saloon was the BE model restarted the VIN numbers from zero.
If that isn't confusing enough, the engines seem a bit more mysterious. The first run had no letter designation. From 1960 they used BA and from 1963 up to final production they used DA. Don't know what happened to CA. This may match with the different engine configurations available during the Fiat 600 production run. The engines seem to have been sequentially number from the beginning.
So, I guess, your car has a late production DA motor (589941) in a late production BE series saloon (VIN 162336). When ordering parts in the day you quoted the number listed next to the Ricambio (spares) stamp 162336.
Not sure what the BY45 or maybe 8Y45 nor what the homologation number means. I'll keep looking around.
Chris