The modern sense of "common, current" is quite often found in Shakespeare. Indeed, its use in Henry V (IV.iii.52) - "Familiar in his mouth as household words" - is the first recorded use of the word in its modern sense.
But other uses are not so familiar. Several reflect the word's origins in Latin familia "family". When it first came into English, in the 14th century, it meant "pertaining to one's family", and thus "intimate". By extension, it was applied to household things, such as animals and food.
So, when Evans describes the louse as "a familiar beast to man" (The Merry Wives of Windsor, I.i.18) he doesn't mean that people readily recognise it, but that it is domesticated. When Iago tells Cassio "good wine is a good familiar creature if it be well used" (Othello, II.iii.300), he doesn't mean that wine is well-known but that it is congenial. And when Falstaff talks of Mistress Ford's "familiar style" (The Merry Wives of Windsor, I.iii.42) he is talking about (what he imagines to be) her welcoming manner.