Two seperate statements. The second, as a weilder, is refering to his Ring I believe. But he is not a servant to the Ring, indeed, he is not a servant to any but Manwe and Eru.
Manwe, the Valar that is in control of Middle-Earth, was well known as the "Sky God" of the Pantheon that the Valar comprised. His symbol were the Great Eagles and his domain the sky and wind.
Eru (the Creator of all), on the other hand, would not have been recognized by many in Middle-Earth, but surely would have been known to the Maiar. Since Gandalf and the Balrog were both Maiar, they would have understood a reference to Eru. The spirit of Eru, called the Secret Fire in the Silmarillion I believe, is what gave life to all living things in Middle-Earth.
It's not hard, therefore, see how a comparison could be made between "The Secret Fire" of Middle-Earth and the Holy Spirit of our world.
Love is a burnin' thing, And it makes a fiery ring
Bound by wild desire -- I fell into a ring of fire.
I fell into a burnin' ring of fire
-- I went down, down, down And the flames went higher,
And it burns, burn, burns, The ring of fire, the ring of fire.
The taste of love is sweet When hearts like ours meet.
I fell for you like a child -- Oh, but the fire ran wild.
I fell into a burnin' ring of fire
-- I went down, down, down And the flames went higher,
And it burns, burn, burns, The ring of fire, the ring of fire.
I fell into a burnin' ring of fire
-- I went down, down, down, And the flames went higher,
And it burns, burn, burns, The ring of fir, the ring of fire.
The ring of fire (and Fade)
Here is where Tolkien mentions the secret fire the first time in The Silmarillion(Near the end of the fourth paragraph of the "Ainulindale":
Never since have the Ainur made any music like to this music, thought it has been said that a greater still shall be made before Iluvatar by the choirs of the Ainur and the Children of Iluvatar after the end of days. Then the themes of Iluvatar shall be played aright, and take Being in the moment of their utterance, for all shall then understand fully his intent in their part, and each shall know the comprehension of each, and Iluvatar shall give to their thoughts the secret fire, being well pleased.
So it seems that the secret fire has something to do with knowledge, which is also one of the ministries of the Holy Spirit.
Wielder of the Flame of Anor is interesting as well. I am not so sure if it refers to the Ring of Fire or not. Anor was another name for Minas Anor (Tower of the Sun), which was changed to Minas Tirith after the Nazgul had taken over Minas Ithil and changed it to Minas Morgul.