Yes, backlink age does matter in 2 very important ways.
1. Aged backlinks pass more link juice in the form of earned trust. Link juice is comprised of 3 values: Relevancy, PageRank, and Trust. Trust is earned over time, so aged backlinks have earned more Trust by having stood the test of time.
2. Since it takes time to earn truly natural organic backlinks, link juice from pages that have been online longer tending to have more PageRank because they have existed longer and had more time to earn PageRank.
Let me point out that technically the “age” of the backlink is not what drives the value of the backlink, but “age” is a prerequisite to those factors, particularly to the Trust factor.