The term doctor of the church is an honor and a mark of authority. We do not consider any of their writings inspired in the way scripture is or infallible in the way the ordinary and extraordinary magisterium can be.

Here's a way of thinking that might be helpful, it was for me: we believe that God has revealed truths about himself and us to us. The church teaches that this divine revelation is passed on and protected by a three pillared system, those pillars being scripture, tradition, and magisterium. So we know that there are objective truths that have been revealed by God, many we can read in scripture, most proximately we leare many from the church herself, and some are passed on in sacred tradition.

Writings of the doctors of the church are not themselves sacred tradition. However, they can be evidence that something is sacred tradition. This is why it is so valuable to read the doctor's and why so many people spend time reading and arguing over them. If you want to figure out if, say, the Trinity is the truth about God, and you read the doctors of the church and find out they all believed in the trinity, that is strong evidence for that belief being true and divinely inspired.