Yes, all those who celebrate Christmas and include Santa in that celebration have a false Jesus. Christ would never endorse a false god.
Exodus 34:14 "14 for you shall not worship any other god, for the LORD, whose name is Jealous, is a jealous God--"
When someone yearns for there to be a Santa, so much so that they look into the skies to pretend he is coming, teaching it to their children and even signing the gifts in Santa's name, leaving out treats for him, have made him a god in their hearts.
Most Christians in our country will not spend as much time talking about Christ as they do about Santa, in fact they almost never teach about Christ but will spend hours talking about Santa and his miracles that never really happen, but everyone wishes would happen.
Matthew 5:28 "28 But I tell you that anyone who looks at a woman to lust after her has already committed adultery with her in his heart."
If God will call it the sin of adultery to so much as wish to be with another woman, how much more He is angered when someone yearns after Santa as their hope and their joy.