Activated ability that share a name
When a card refers to it's own name in text, it isn't actually referring to cards with that name; it's referring to itself.

So Vector Asp's activated ability actually means: "{B}: This Card gains infect until end of turn". You can attack with both Vector Asps, and activate each ability separately, to give one, both or neither of them infect.

Some cards will refer to other cards with a certain name, but this is usually formatted as "... a card named ~" (eg. Accumulated Knowledge).