It's explicitly said that the reason Saber has Invisible Air is to conceal her identity, that is, her sword, not even herself, is so identifiable people would know her by looking at it. They don't necessarily know the specific form of the vessel the servant is in though, cause if they knew it would be useless to call themselves by their class instead of proper name.
The more famous a myth or legend is and the more closely related the HS are, the more they seem to know about each other. Li Shuwen can recognize Kanshou and Bakuya for example, and even says they are not the originals. (K&B being originally Gan Jiang and Mo Ye, a chinese legend)
A basic knowledge of who X is though seems to be imparted in every servant. If someone says Cu Chulainn, they will know him, but don't except a non-Celtic hero to know Gae Bolg just by seeing it.
In FGO they have a Watsonian explanation for why they have the same memories - they have a "copy" of the Saint Graph of the servant as data. (Don't know if you've played the chinese lost belt, but this detail plays a part in one event there) It was in a computer before, in Chaldea, now it is a suitcase. This isn't a "true/normal" summon in a proper sense, though. Normally, you don't have the Saint Graph itself, so you can't exactly choose the SPECIFIC servant to be summoned, so normally it wouldn't be like that.
But, as TypeOU said above, there are some inconsistencies in this part.