Yes! And, in that sense, "régi" can be used with persons, as well. For example, "my old boyfriend" would be "a régi barátom". And it does not mean that he is actually old. Only that he used to be my boyfriend.
Or "we are old acquaintances" would mean "régi ismerősök vagyunk". That is, we have known each other for quite a while. But it does not mean that we are actually old.
Great explanation! But I think I can observe here that "régi" refers not to the persons here, but the relations. The friendship or acquintance is "régi", even if the person is "fiatal" (young) or "öreg" (old). You may want to make a little analysis what régi refers to in uncertain structures and though I think it is not completely universal, you can use "öreg" to the persons and "régi" to relations with them.
Nevertheless it is not something I read or learnt, it is just my observation right now so it may be wrong.
Previously I mentioned that "öreg" may express a personification. An example: a willow tree could be either "régi" or "öreg" but mostly the latter. As a living thing the "öreg" won't express the personification like in "Old Man Willow" by Tolkien. But old bridges, towers, etc. usually called "régi híd", "régi torony" in general. When it comes to a special landmark, a popular old building, it usually but not necessarily changes to "öreg híd" or "öreg torony" (latter is often in form of "öregtorony" though I think that is a special kind of fortress towers, too). "Öreg híd" and "öreg torony" almost always demonstrate fondling love and care from the local people. Perhaps we could translate them as "Ole Man Bridge" or "Ole Man Tower", but in this context "Ole Man" usually translates to Hungarian as "Apó". (That is a very old and beloved member of the family, or a revered and very old person or simply great reverence. (The Polish general Józef Bem earned his nickname in 1848-49 "Bem apó" though he wasn't that old then—but very small in stature.)
Mind that the accents have very important role in Hungarian and don't let them foul yourself. The "reg" is not a root in these words, it just happened to look somewhat similar. "Reg" is an archaic poetic form (or thought to be) of "morning" while "rég" is "in the past". Confusing the two is almost similar like confusing "morning" and "mourning".
interesting question.i've just learned,that "REG" meant "warm,hot,warmth" in ancient hungarian.actually you can relate it to morning("REGgel")-like,after a cold night comes the more warm morning with sunshine...but i don't see any connection between this and "öREG,RÉGi"(they mean "old",as you surely know).still there can be,i didn't find anything useful.