In most cases, DVD-R discs all have the drive make/model written to them. Some drives also write a decent amount of information to DVD+R discs [but not always]. As for CD-R, I do not think that such information is recorded–at least I’ve never seen the data displayed.
ImgBurn can show the “recorded with” information on DVD+R if the drive wrote such information [and it is usable], and ImgBurn and Nero CD-DVD Speed can show the “recorded with” information on DVD-R.
Screenshot of DVD+R in ImgBurn, DVD+R DL in ImgBurn, DVD-R in CD Speed, DVD-R in ImgBurn. Drive information is highlighted in yellow. Notice that “garbage data” may sometimes be written [as in the last screenshot]. The drive used for all discs was [obviously, hehe] the LG [aka HL-DT-ST] GSA-E40L.
Note that some drives may write a generic label – such as just “ATAPI” or the name of the chipset used or the brand name – instead of a combination of make and model.