The person saying athkaar should concentrate on the meanings of what he is saying, such that he would enjoy their benefit, reward and protection, as is indicated in revealed texts. As for saying the morning and evening athkaar while listening to the Qur’aan, this is totally inappropriate, as he is not actually listening to the Qur’aan, and thus he is going against the commandment,
(وَإِذَا قُرِئَ القُرْآنُ فَاسْتَمِعُوا لَهُ وَأَنْصِتُوا)
“… so when the Qur’aan is recited, then listen to it and pay attention.” (Al-A’raaf 204)
This person is not paying attention while saying the morning and evening athkaar, and is thereby going against what is stated in the aayah. Thus, he will not benefit from saying these athkaar, since he has not focused on them nor contemplated their meanings. He may receive the reward for simply saying them, as was the opinion held by Ibn Hajar, as the revealed texts specify that whoever says such-and-such gets such-and-such a reward. This reward is achieved by the mere virtue of the statement, whereas understanding its meanings and contemplating them is something extra in the opinion of Ibn Hajar. Other scholars hold that simply pronouncing a statement with the tongue is insufficient, until one recalls its meaning in his heart and contemplates it, whether it is from the Qur’aan or athkaar.