Jaqen H'ghar might have been referring to R'hllor when he mentions the "Red God", but later on in the series, a different god is revealed to be the patron of Jaqen's order, the God of Many Faces.
It could be that acolytes assume that R'hllor is an aspect of their god, but it's never explicitly mentioned, and there is a temple to both R'hllor and the Many-Faced God in the city of Braavos.
For Syrio Forel, it could be that he means it less from a religious point of view, and more from a martial point of view - "don't die today
" sounds like a pragmatic soldier's prayer.
R'hllor is supposed to be the antithesis of death - he's the god of life and fire, but that doesn't mean Syrio worshiped him.
scifi.stackexchange.com/users/2417/Sorcerer13
Sorcerer13 is right.
You will understand more fully after reading A Feast for Crows .
Syrio Forel's "religion" doesn't get explicity revealed, though that's what makes the plot so good.
If there were an IRL equivalent, I would say its pretty Buddhist (no flamers please, just my opinion!)
Jaqen H'ghar's "seems" to be a worshipper of the Many-Faced God of Braavos (though never explicity stated), he does give her the coin and the phrase "Valar Morghulis"-- actions which will reveal themselves in later books/episodes.
Melisandre is a priestess of R'hllor, which to me seems pretty unbending in its lack of tolerance of other gods/deities.
scifi.stackexchange.com/users/6212/Lyoness
After thinking on this question for a few days, I think it's worth pointing out that Jaqen H'ghar saying that Arya "stole three deaths from the red god" might refer to the fact that Arya saved them from death by fire, which is what Melisandre's god R'hllor is typically associated with.
The other answers have done a great job pointing out that they do not worship the same god (assuming they are not the same god, or that those gods even exist in the first place).
scifi.stackexchange.com/users/6397/Kyle S.
It was my understanding that by "the red god", he meant the fire that Arya saved him from.
scifi.stackexchange.com/users/6547/Trev
According to GRRM, Jaqen was a bit influenced by his recent experience...
Jaqen refers to the Red God, and elsewhere to the god of fire.
Is he referring to R'hllor?
When we see Arya being educated by the Faceless Men, R'hllor doesn't seem to be particularly important to them.
(George thinks for a moment)
Well, remember when Jaqen names him: he had very nearly burned to death recently... ASSHAI.COM INTERVIEW IN BARCELONA - July 28, 2012 but the Many-Faced God is called because he represents all the other gods.
There are roughly 30 statues of other gods in the House of Black and White.
"Then you have come to the wrong place.
It is not for you to say who shall live and who shall die.
That gift belongs to Him of Many Faces.
We are but his servants, sworn to do his will.
"
"
Oh.
"
Arya glanced at the statues that stood along the walls, candles glimmering round their feet. "
Which god is he?
" "Why, all of them," said the priest in black and white.
A Feast for Crows - Arya II While on the other hand it seems the worshippers of R'hollor are intolerant of other religions, as evident by Mellisandre burning the effigies of the Seven.
The red woman walked round the fire three times, praying once in the speech of Asshai, once in High Valyrian, and once in the Common Tongue.
Davos understood only the last. "
R'hllor, come to us in our darkness," she called. "
Lord of Light, we offer you these false gods , these seven who are one, and him the enemy.
Take them and cast your light upon us, for the night is dark and full of terrors.
"
A Clash of Kings - Davos
I
So in conclusion...
They do not worship the same gods.
scifi.stackexchange.com/users/57310/Skooba
here are the answers: Jaqen:
In the temple of the Many Faced God, there are statues of many different gods.
The priests believe that these 10 or so gods are really just different faces of the same god.
Thus, someone from the South of Westeros might call the Many-Faced-God, "The Stranger."
Someone from Asshai would most likely call the God-of-Many-Faces, "The Red God.
" To them, they are the same thing.
So if Jaqen if from Asshai, then the Red God and the Many-Faced-God are interchangeable names.
Syrio:
In regards to Syrio, it seems that everyone from Braavos has a healthy respect for the Faceless men (and more than a little fear of them).
For example, Jaqen tells Arya to give the coin to anyone from Braavos, not just a religious person.
Syrio just seems to be relaying the philosophy that everyone from Braavos grew up with.
scifi.stackexchange.com/users/7790/Scott
Jaqen serves the Many Faced God.
He and two companions were saved from a terrible death by fire by Arya.
He simply mentions the "Red God."
The Red Priests and their god are evil.
The Many Faced God gives the gift of death to those who are in too much pain(physically/emotionally).
It is not taken lightly.
Those who serve the Many Faced God also mete out death to individuals:
that take advantage of people, those deemed evil, etc.
But in no way is R'hollor is not interchangeable with the Many Faced God.
Syrio certainly knew of many gods.
However, was his god the Many Faced God?
I don't think so.
I think he told Arya that the only god is death to help her with her "Water Dancing."
"Just so!"
scifi.stackexchange.com/users/29006/driftwoodms14
Melisandre worships R'hllor, the Lord of Light and god of fire. That much is clear. According to the  Game of Thrones wiki , R'hllor is also known as the Red God.  After Arya saves Jaqen H'ghar from death, he says she stole three deaths from the "Red God." Is that a reference to R'hllor?  Also Syrio Forel, the Braavosi water dancer tells Arya that there is only one god - the god of death. Is he also talking about the same god?
Do Syrio Forel, Jaqen H'ghar and Melisandre all follow the same religion?
scifi.stackexchange.com
<game-of-thrones><a-song-of-ice-and-fire>
scifi.stackexchange.com/questions/16714
scifi.stackexchange.com/users/3442/Royal Flush
Jaqen H'ghar might have been referring to R'hllor when he mentions the "Red God", but later on in the series, a different god is revealed to be the patron of Jaqen's order, the God of Many Faces. A different God is revealed later in the series but it is very possibly a reference to R’hllor. 
A different God is revealed later in the series but it is very possibly a reference to R’hllor. 