And this can be refined as "régi" is not necessarily referring to age. In that sense, it would mean "from old/previous times". And, also, "öreg" can be used for objects, especially if you have a sentimental connection to them.
"az öreg autóm" - my old/aged car
"a régi autóm" - my old/previous car
This car could have been brand new, and still very young when I sold it. But it was a car I had a long time ago. A previous car.
Similarly, for persons:
"az öreg barátom" - a friend of mine who is old
"a régi barátom" - a long-time friend of mine