Seraph and Seraphim are generic, but SwLC is a unique character. It is the specific Seraph which held the Live Coal.
On the question at hand, this has always been ruled via Angel in the Path and Angel's Sword granting NoNB that sharing the reference also qualifies:
If an enhancement is played on a character, and the title of the enhancement contains the name of that character it gets Name on Name bonus. Also, if the character and the enhancement have the exact same scripture reference they get name on name bonus.
Examples:
Moses + Covenant of Moses (same name)
Job + Job's Faith (same name)
Huge Egyptian + Egyptian Spear (same scripture reference)
Woman of Thebez + Stone of Thebez (same scripture reference)
High Priest Caiaphas and High Priest's Plot do not share the same name (Caiaphas) or scripture reference so they do not get Name on Name bonus.
By this definition,
Seraph w/LC and
Live Coal have NoNB, since they both have a reference of Isaiah 6:6-7. This has been ruled very consistently through the years.
However, the definition in the new REG does not have anything beyond matching titles:
Name-on-name bonus is an advanced rule (see Name-on-Name Bonus in the rulebook). When this
rule is in effect, an enhancement card with a person’s name in the title is worth double its abilities
(*/*) when played on a character card that has the same specific person’s name in the title. Play
name-on-name bonus as “person's-name-on-same-person” bonus, because there are examples in
the Bible where the same person has multiple names that are reflected in Redemption. Name-on-name bonus does not affect nor is it affected by the special abilities on cards. Characters must be unique; generic characters cannot be used for name-on-name bonus.
I would like to say go with the original, long-standing wording, but we need clarification on whether this changed in the latest rules update.