I wonder when the author of this statement last read the New Testament. Matthew 2:1-2 reads, "In the time of King Herod, after Jesus was born in Bethlehem of Judea, wise men from the East came to Jerusalem, asking, 'Where is the child who has been born king of the Jews? For we observed his star at its rising, and have come to pay him homage.'"
The wise men came to visit Jesus after he was born, not on the day of his birth.
For Christians, doesn't Christmas mark the birth of Christ the king, not a "new King"? Who are they referring to? Jesus or Donald Trump?
And finally, this statement speaks from a Christian perspective - there's no differentiation between the speaker and the Christian message - "a Savior who would offer the promise of salvation to all mankind." Jesus is unequivocally the Savior. While the message starts, "Merry Christmas to all," the Republicans are speaking to their fellow Christians in this message, not to all Americans.
So what wishes did the Republican Party send about Hanukkah this year?
This message differentiates the speaker(s) from those receiving the wishes. They speak to "our Jewish friends." The statement is addressed to Jews specifically, without the false universalism of the Christmas wishes. I find it creepy to be addressed as "our Jewish friends," with the clear indication that "we," the Republican party, has Jewish friends - but not that the "we" includes Jews (although of course there are plenty of Jewish Republicans).
And the Democrats? Both messages are addressed to those celebrating. The Democratic message doesn't send Christmas wishes to "all," unlike the Republican message. Jews are not "our Jewish friends"; they, like Christians, are those who celebrate their specific holiday.