An example of a moggy is a cat that is not a purebred.
Original sense, early 19th century, is a term of affection for a calf or cow, which may have been transfered to cats under urbanization. Later 19th century meaning of “untidy woman, slattern". Alternatively, in Wigan, moggy traditionally applied to mice, not cats, and a cat was hence a moggy catcher, which may have been abbreviated to moggy.
Apparently not an abbreviation of similar-seeming mongrel, though perhaps from similar Old English/Proto-Germanic source; see mongrel for details.