In Psalm 103:8 we read:
אֶ֖רֶךְ אַפַּ֣יִם וְרַב־חָֽסֶד
Why is אַפַּ֣יִם (anger) in its dual form?
We need to explore the word אף more closely. It has two general meanings: nose, face, nostril (Genesis 2:7, Proverbs 30:33, Genesis 28:12, Samuel I 25:23, Chronicles II 7:3) and anger (Deuteronomy 29:27, Proverbs 30:33, Daniel 11:20, Zephaniah 2:2, Genesis 30:2).
Why this is the case is left to your interpretation. On this point I agree with this answer - that the nose/nostrils seem to be an expressive feature when angry, and for rarely anything else. So it makes sense that there is a strong association between the two. This might not be the case however, since there are occasions where "face" means "anger" (Ezekiel 15:7), in which case maybe the explanation centres around a connection to the face and not directly to anger.
The dual form is not exclusive to one meaning - both meanings have singular and plural examples. In the incredibly vast majority of cases taking the meaning "anger" the word takes the singular form (חרון אף, ויחר אפו, ויחר אף, וחרה אפי). If so, it seems to be to make great sense in describing forbearance and patience that the double form is used for extra emphasis - to express that the patience extends to an even greater anger. Of course, the dual form isn't always used, and there might be literary/poetic preference in particular cases. I.e. Sometimes a particular form might sound better. Although this might be another reason to use a plural (and in particular the double) form in other cases.
So to recap, we've proposed two reasons - the first is emphasis, and the second is literary form/rhythm.