D

$A.\because \angle DBC=\angle DAC$，
$\therefore \angle BOC=\angle AOD$，
$\therefore \triangle AOD$∽$\triangle BOC$，
$\therefore \frac{AO}{BO}=\frac{OD}{OC}$，
$\therefore \frac{AO}{OD}=\frac{BO}{OC}$，
又$\because \angle COD=\angle AOB$，
$\therefore \triangle AOB$∽$\triangle COD$，故选项$A$不符合题意；
$B.\because \angle ACD=\angle BAC$，$\angle AOB=\angle COD$，
$\therefore \triangle AOB$∽$\triangle COD$，故选项$B$不符合题意；
$C.\because \angle COD=\angle AOB$，$\frac{OA}{OD}=\frac{OB}{OC}$，
$\therefore \triangle AOB$∽$\triangle COD$，故选项$C$不符合题意；
$D$.条件$\angle AOB=\angle COD$，$\frac{OA}{OC}=\frac{OD}{OB}$，无法证明$\triangle AOB$∽$\triangle COD$，故选项$D$符合题意.
故选：$D$.