Yes and no.
There are two types of mod for the 360, maybe three but nobody really cares about shared live profiles and related stuff there these days (shared profiles aside it was all very minor and does not add much to things).
DVD mod.
This allows you to play as it was 1:1 DVDs from a compatible region if you burn them. There are ones that replace the drive and take in a USB there too. The drive replacement side of things is obvious but the plain DVD mod is not.
MS did also release an update for some later games that rewrote the DVD firmware to allow it to support slightly larger games (see XGD3) that might mean both the DVD mod and its abilities to play older burned games is gone and you will need to find the key and flash it again, as well as maybe reburn some of those discs after being fixed. There are also some other games that do other things here (see AP2.5) but we will skip that for now. Equally some of the really old stuff might not support the newer things (its internal security blocking them as bad when in fact they are just newer than it knows about).
You can check the discs if you want with a PC tool called ABGX360 and see what that says.
If you tear into the drive in some models there will be signs of wires and other things, or replacement PCBs. Others will appear as stock drives.
https://gbatemp.net/threads/guide-how-to-hack-your-360.334203/ has some more there, and also for what follows, and what you might need to have to attempt it.
It playing a copied DVD is a fairly decent giveaway of a mod, however there are also plenty of reasons why some of those existing discs might not work so I am not 100% pending what kernel it is on, what drive type it is (you can tell from external clues) and what ABGX360 says about the discs in question. I am not sure I have a reliable way of telling with it in the system other than it reading a copied disc. Outside the system with a flashing setup then maybe, though at that point...
Dash mods and other things.
Cheats, homebrew, xbla, dlc, modded versions of all of said same, region free, USB loading, hard drive loading, any replacement hard drive you can convince to be powered by it, software USB mouse-keyboard adapters... does tend to come at the cost of online play though* unless you do system link over the internet (which you can do with a modded box and a stock box as the ping checks are only for the host I think it was). Amusingly enough does not allow copied DVDs to inherently work (enough of the security being in the DVD drive means it can not command it to ignore it from the system itself), though if it is modded you can read out the various keys and thus decrypt the DVD keys which is usually 99% of the annoyances with DVD modding, not sure what went with the ability to flash from the modded dash.
There are some things you can try like powering on with eject and you might see the xell/xellous bootup screen vs the flashy normal power on. Equally if you can see inside the case then the wires needed to do either a JTAG mod or a RGH mod will be obvious. This sort of mod is far rarer than a plain DVD mod.
*dvd mods can do online, in the past there were banwaves but I doubt we will see another. For RGH there is also dual NAND option but it is very annoying and tedious to do.
