My take on this, and to denote the tones herein:
"Happy Easter, Happy Passover, Happy Christmas, Happy St. Valentine's day, happy St. Patrick's day, etc.." is a well wishing on the others for a pleasant holiday, a pleasant time off, go do what you want, how you want, and with whom you want. No direct "religious" connotations attached.
(Ask any kid in school about Easter, and it's a day off, maybe two, maybe a week (2 if they're orthodox). chocolate, chocolate, that egg smashing game, A bog meal with ham or lamb (or Brisket, Matzos, Kishka, latkes, gefilte fish), and maybe being dragged to church on Sunday..)
"He is risen, He died for you, The one on the middle cross, Happy birthday Jesus, All praise Allah, And the Israelites started their journey, Moses descended from.." all have a religious connotation. they are all implying some recognizable aspect of a known religion, and are recognized as a celebration of that religion. Despite them all having an equal "happy [tag-line]" connotation.
And, different places put a different emphasis on these days & saying. Your town may shout down for this particular week-end, mine doesn't. Mine will shut down for another week-end (Festival du Poutine! and it religious to some!)
So, you wanna wish me a happy Easter, happy holidays? Fine, same to you!
You wanna tell me how happy you are that he's risen, that your son gave you a gold plates menorah with self lighting candles? Uh, yeah, OK... good for you?