In Psalm 103:8 we read:
אֶ֖רֶךְ אַפַּ֣יִם וְרַב־חָֽסֶד
Why is אַפַּ֣יִם (anger) in its dual form?
This is the same phrase that appears in Ex 34:6, the interaction between God and Moshe after the golden calf. In both places אֶ֖רֶךְ אַפַּ֣יִם is sometimes translated "slow to anger" (though there is no infinitive verb there) and sometimes "long-suffering".
אֶ֖רֶךְ is Strong's H750, "prolong, lengthen, draw out" (etc). אַפַּ֣יִם is H639. The lexicon gives two usages here: "anger" (derived from H599) and "nostril, nose, face" (specific to this form of the word, H639). While many uses of the word in Tanakh are about faces rather than noses, apparently not all are.
What does anger have to do with the face or nostrils? I think the connection is that one's face flushes and nostrils flare when one is extremely angry. The text isn't talking about run-of-the-mill anger that you'll get over in an hour or two; it's talking about divine anger, anger that leads God to almost wipe out whole peoples sometimes, anger that is being brought up right after the incident of the golden calf and Moshe's plea for divine mercy.
The kind of anger that makes one's nostrils flare.
Why the dual form? We have two nostrils. In Hebrew the dual form is always used for paired body parts -- yadayim (hands), reglayim (feet), einayim (eyes), and so on.
This answer is my own reasoning based on the sources cited. I don't have a commentary or lexicon that makes this connection explicit. According to @Mike in the comments (thanks!), related idioms are discussed in Beginning Biblical Hebrew by Mark D. Futato.
Please note: This answer was written for a neutral, academic audience and is not intended to be interpreted in the context of a religious belief or doctrine.
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.