①$\triangle ADB$≌$\triangle CBD$，

理由是：$\because AB$∥$CD,AD$∥$BC$，

$\therefore \angle ADB=\angle CBD$，$\angle CDB=\angle ABD$，

$\because BD=BD$，

$\therefore \triangle ADB$≌$\triangle CBD.\left(ASA\right).$

②$\triangle ADE$≌$\triangle CBF$，

理由是：$\because AB$∥$CD,AD$∥$BC$，

$\therefore$四边形$ABCD$是平行四边形，

$\therefore AD=BC$，$CD=AB$，

$\because CF$∥$AE$，

$\therefore \angle AEF=\angle CFB$，

$\because \angle ADB=\angle CBD$，

$\therefore \triangle ADE$≌$\triangle CBF\left(AAS\right).$

$\therefore DE=BF$，

$\because BD=BD$，

$\therefore DF=BE$，

$\because AB=CD$，$\angle ABD=\angle CDB$，

$\therefore$③$\triangle DCF$≌$\triangle BAE$，

$\therefore$共$3$对.

故选$A$.