A molecule (X) with following structure under mild acidic condition...

Molecule $(X)$ is an ether with two enol ether functional groups attached to the central oxygen atom. On mild acidic hydrolysis, an enol ether cleaves to give a carbonyl compound and an alcohol.

The $1$-propenyl side $(CH_3-CH=CH-O-)$ hydrolyses to propanal, $CH_3CH_2CHO$.
The isopropenyl side $(-O-C(CH_3)=CH_2)$ hydrolyses to the enol of acetone, which tautomerises to acetone, $CH_3COCH_3$.

Therefore, $(Y)$ and $(Z)$ are propanal and acetone.

Statement A: Both $(Y)$ and $(Z)$ have the same molar mass.
Propanal and acetone are functional isomers with the same molecular formula $C_3H_6O$ and hence the same molar mass $(58\text{ g mol}^{-1})$.
Statement A is true.

Statement B: $(Y)$ and $(Z)$ can be distinguished by $NaHCO_3$.
$NaHCO_3$ reacts only with carboxylic acids (releasing $CO_2$). Neither an aldehyde nor a ketone reacts with $NaHCO_3$, so it cannot distinguish between them.
Statement B is false.

Statement C: $(Y)$ and $(Z)$ react with HCN at the same rate.
In nucleophilic addition to carbonyls, aldehydes react faster than ketones due to lower steric hindrance and a more electrophilic carbonyl carbon (fewer $+I$ alkyl groups). So propanal reacts faster than acetone.
Statement C is false.

Statement D: $(Y)$ and $(Z)$ undergo addition reaction with $2,4$-DNP.
Both aldehydes and ketones react with $2,4$-DNP (Brady's reagent) by nucleophilic addition followed by loss of water to give coloured $2,4$-dinitrophenylhydrazone precipitates.
Statement D is true.

Hence, the correct statements are A and D only, which corresponds to option $(4)$.