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Yes, indeed! There are very old traditions about this in both the eastern and the western churches. There are many religious orders that wear unique styles of clothing called "habits." Examples would be Benedictines, Dominicans, Franciscans, and others. Just do an online search for religious habits and you will find many of them pictured. In some cases there are even miniature items that resemble pieces of the full-sized habits for lay people to wear who may have an relationship to a particular religious order or perhaps who wish to learn from the spirituality of a particular order or its founder.
In addition, there are other clothing-related traditions among Christians. Mormons wear a type of undergarment that has religious significance. And some Christian sects put restrictions on what their members can wear (such as forbidding short dresses or short pants).
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Biblically, the 'clothing' that the Christian is to wear is Christ:
"(C)lothe yourselves with the Lord Jesus Christ, and do not think about how to gratify the desires of the sinful nature." (Romans 13:14 NIV)
The garment that we wear is to be the righteous likeness and character of Christ:
"(P)ut on the new self, created to be like God in true righteousness and holiness." (Ephesians 4:24 NIV)
"(C)lothe yourselves with compassion, kindness, humility, gentleness, and patience." (Colossians 3:12 NIV)