Is there an American English equivalent for the British "moggie" for a non-purebred cat?

Looks like domestic short-haired cat and domestic long-haired cat are standard in American English.

Wikipedia:

A domestic short-haired cat is a cat of mixed ancestry – thus not belonging to any particular recognized cat breed – possessing a coat of short fur. In British English, they are often referred to as moggies.

If you want simple, house cat and alley cat often connote mixed breed, especially the latter, but are technically terms describing where the cat lives, not its ancestry. Nonetheless, house cat is used by some cat organizations (source) to refer to mixed-breed cats of both long-hair and short-hair varities.

I'll go with mongrel cat.

alley cat: a homeless, mongrel cat Your Dictionary

Thomas, a 10-year-old moggie (a British nickname for a housecat or mongrel cat), claims to be Britain's fattest cat at 28 pounds. Books Google

I hadn't realized it, but a smorgasbord of choices exists, with one or another better or worse, depending on your specific reference cat-sense. The following are some, omitting 'moggie' and others already mentioned in earlier answers. If the sense desired includes a denoted (rather than connoted) 'mixed breed', 'housecat' (already detailed in another answer) in use is probably closest to the use of the British 'moggie' (other than 'mog', also British).

Obsolete, archaic or rare:

• bad (obs.)
• gib, or gib-cat (obs. gib, arch. and dial. gib-cat)
• mewer (rare)
• mewler (obs., rare)
• Tibert (arch.)
• miauler (rare)

Others:

• baudrons (Sc.)
• puss-cat (Am. or Brit., nursery or colloq.)
• puss (colloq.)
• grimalkin (esp. an old she-cat)
• miaower
• pussycat (colloq., orig. nursery)
• pussy (nursery and colloq.)
• tigerkin
• pussums (colloq.)
• mog (Brit. colloq.)

(All terms and designations [obsolete, archaic, rare, etc.] from the OED Historical Thesaurus.)

The 'tigerkin' choice seems best to me, and avoids denigrating the cat while at the same time suggesting in a positive way that the cat is a mixture of breeds.

American here, but which type of moggy?

shorthair moggy's American equivalents:

• Domestic cat
• Mongrel cat
• Mutt cat
• Mixling cat
• Domestic mixed-breed
• Domestic mix-bred
• Domestic mix
• Mix breed
• Mix-bred
• Mix
• House cat, Alley cat
• unknown-breed cat
• Dsh/Domestic shorthair
• Regular house cat
• Common shorthair
• Normal shorthair
• Regular shorthair
• Domestic house cat
• Regular cat
• Common cat
• Common house cat
• Tyke cat
• Crossbred cat
• Outcross cat
• Halfling cat
• Random-bred cat
• Rando cat
• gatocat

(For this one, I'll go with gatocat.)

Mediumhair moggy's American equivalents:

• Dmh/Domestic mediumhair
• regular mediumhair
• normal mediumhair
• common mediumhair
• somali mix?
• classic angora
• vixen-tail cat

(for this one, I'll go with vixen-tail.)

Longhair moggy's American equivalents

• Dlh/domestic longhair
• Regular longhair
• Normal longhair
• Common longhair
• long muzzle persian
• doll faced persian
• doll face

(for this one, I'll go with doll face.)

These are all examples, in my vocab. I've been around people who say these phases here in Cali, and decided to put them in my vocab.