Channuka is only a “modern iteration of christmas” for those who choose to celebrate it that way. However, the lighting of the Menorah does date from very early times, as reflected in discussions dating back to the talmud. True, the Jews didn’t show up at the Temple Mount to light the Menorah and give presents because that was never ordained as the way of celebrating this holiday.
Karaites are a very small minority of Jews (or believed to be Jews) who reject the Oral Law of the Torah (the Mishna and the Talmud, eg.). Traditional Judaism therefore regards their practices as something other than Judaism. Consequently, a Karaite Rabbi is at best a rather dubious source for information on Judaism, even in a historical perspective, since he rejects most of the sources we rely on.
Bottom line: the lighting of the Menorah in Jewish homes to commemorate the miraculous events of the Jewish rebellion against their foreign oppressors and the aftermath, has been around for about 2,000 years. And yes, there are even some customs, predating Christmas, regarding gift-giving, primarily in the form of small amounts of money to children, known as “chanuka gelt”.
Chanukah isn’t a modern iteration of Christmas; Christmas is a modern iteration of Chanukah, in a sense. Chanukah includes Jesus’ celebrated birthday, the 25th of Kislev, when the Spirit of God dwelt once again in the Temple of Israel.