The word "register" when referring to music means a particular range of notes. So a "deep register" is a range of notes at a low pitch.

Often, a musical instrument can produce a low note if it's played in a certain way, and a high note--perhaps an octave higher--if it's played almost the same way, but with some slight difference in technique or strength. A singing voice will often have several "registers." For example, male singers will sometime sing an octave higher using what is sometimes called "falsetto." That would be their "upper register." Conversely--I'm not sure about this, maybe someone else can help me out--I think female sopranos naturally sing in an upper register, and only rarely sing in their "lower register."

Thanks a lot!

"Sung in a deep register" would almost always be talking about using low notes. A use of it to mean serious would be be just a poetic analogy that a particular author might use, but would not be common. It is true thought that "deep" commonly means "serious" or "insightful".