Both are right, but in the context I think you mean, My wife is almost always preferred. My wife, his wife, (heteronormativity off) her wife, your wife, etc. is the standard use when you want to say something about your wife. This holds for many nouns, not just wife. In other situations "the wife" would be totally normal such as 'The wife of a dead man is called a widow.' However, you are very right that the wife can be idiomatic; it has a very specific meaning with a specific connotation.
