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Slightly surprisingly, Pepys refers to this as being a Tavern House (16/3/1660) but they were more often known as Rhenish Wine Houses.
Rhenish means the Rhineland, which is the home of most German wines, which were highly regarded.
John Taylor, in 1636 lists "foure houses in London that doe sell Rhennish Wine, inhabited only by Dutchmen namely: the Stilliyard (Steelyard, off Thames Street); the Swan in Thames Street; the Swan in Crooked Lane; The Sun at St Mary's Hill".
However, by the 1660s there was also the "old Rhenish Winehouse" somewhere in King Street which seems to have survived at least to the 1720s, and that is is probably the one.
Source: London Signs by B Lillywhite.