A lot of confusion here!

First, this has nothing to do with the difference between a CD-R (recordable) and a pressed CD. Any of the following could apply to either type.

A CD with MP3 files on it is a CD-ROM, which is computer file system on a CD. Computers can read it but many audio CD players can't. The ID tag is part of the MP3 file header. Some players, especially car players, CAN read CD-ROMs with MP3 files on them, because it's a convenient way of getting about 10x as much music on one CD, at a reduced quality level but good enough for listening in the car...

It is possible to burn an Audio CD from MP3 files - that's an Audio CD and not the same as above.

It IS possible to embed text on an Audio CD. CD-TEXT was invented by Sony and is an extension to the Red Book standard. Some CD burning programs support it - you can put track titles, an album title, an ISRC number for each track and several other pieces of information. If such a program can works from MP3 files, it will almost certianly let you copy the MP3 ID tages into the CD-TEXT track titles.
A CD with CD-TEXT on it is playable on any player, even if the player can't read the text.

CDDB is one of several internet databases of album and track titles identified by clues such as track lengths. It's a somewhat hit-and-miss method but it seems to work well enough. It is quite independent of CD-TEXT.