As late as Slott’s Go Down Swinging it seems that Eddie still does not remember that Peter is Spidey, as MJ refrains from using his name or referring to May as “Aunt May” while talking to Peter when Eddie is in the same room. However, during the Cates/Stegman run, it is established that Eddie knows Peter is Spider-Man, is quite friendly with him, and even calls him “Pete.”
I can’t remember if I missed anything from Mike Costa’s run on Venom around the same time as Go Down Swinging, but I’m pretty sure that Eddie never remembers Peter is Spidey during that run as well.
There is an easy explanation since Eddie lent Peter the suit during Go Down Swinging before returning it to Eddie afterwords, which would be that the suit probably remembered once bonding with Peter and then Eddie regained the knowledge after the suit was returned. However I don’t remember any of this being specifically addressed, please let me know if I am missing anything!