It follows 66 Labour MPs voting to extend air strikes to Syria, a policy opposed by party leader Jeremy Corbyn.
Mr Morgan told BBC Wales most party members were "very loyal" to Mr Corbyn and "few" backed wider military action.
But he said most Labour MPs were "not really Corbynistas", so the party was having to deal with a "new phenomenon".
Speaking on the Good Morning Wales radio programme, Mr Morgan said: "In effect we have two Labour parties and this is a bit of a new phenomenon, so we've got to try to let it settle down and work out what's the best way forward."
Mr Morgan called a suggestion by the Labour MP and former minister Frank Field that the party should have two leaders was "quite clever" but "not very workable".
Mr Morgan's comments came as Labour celebrated victory in the Oldham West and Royton by-election.
Deputy party leader Tom Watson called the result "very, very good" for Mr Corbyn, and urged MPs to "swing behind" their leader after a "difficult week".
Labour is effectively two political parties, due to the split between its MPs and wider membership, ex-Labour First Minister Rhodri Morgan has said.