Assalaamu alaykum. We know that anger is bad and that one who avoids being angry for Allaah's sake will be rewarded. My question is the following: suppose that a man gets angry at his driver and intends to rebuke him. Then his wife requests him to avoid being angry for her sake, and he calms down to make her happy, not for Allaah's sake. Is this act Shirk (polytheism) on his part? Will he be punished for it?

All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger.

This is not Shirk at all; suppressing one’s anger is among the good habits, and it is not a condition to have an intention for this, so it does not involve showing off. Ash-Shaatibi said in Al-Muwaafaqaat:

“The scholars agreed that habits do not require an intention, and this is enough to show that intending a worldly benefit does not undermine the acts resulting from it. If we assume that a man got married in order to show off with his marriage, in order to be considered among the chaste people, or for any other intention, then his marriage is still valid, as it is not legislated to have the intention of worship when getting married so that we can say that showing off or seeking a good reputation could undermine it, contrary to acts of worship that are solely intended to glorify Allah.” 

This man is not sinful, but if suppressing his anger was not for the sake of Allah, then he misses the reward for that.

Ibn Rajab said in Jaami’ Al-‘Uloom wal-Hikam:

“The saying of Allah (which means): {No good is there in much of their private conversation, except for those who enjoin charity or that which is right or conciliation between people. And whoever does that seeking means to the approval of Allah - then We are going to give him a great reward.} [Quran 4:114] Allah negated good from many things about which people have private conversation, with the exception of enjoining good, and Allah specifically mentioned charity and reconciling between the people, due to their general benefit. This proves that having private conversations in order to reconcile is good, but as regards getting the reward for it from Allah, then Allah specified this for the one who does it for the sake of Allah…” 

Allah knows best.