In this year is a prepositional phrase with this year as the object. This year by itself is a noun phrase.
Both can be used to place an event within the current year, but this year is much more natural. (I don't think in this year is a hypercorrection.)
In this year, Tommy has grown 12 inches. (odd)
This year, Tommy has grown 12 inches. (ok)
But if we're not talking about something happening but counting the number of days in a particular span of time, the preposition is necessary.
She left him this year. (ok - she left him denotes an event)
How many days are left this year? (very odd - left is an adjective)
How many days are left in this year? (ok)
Expressions telling when are sometimes prepositional phrases (during exams), sometimes adverbs with or without modifiers (later, very soon now), and sometimes noun phrases (next Tuesday, this year).
Happy Christmas is fine in British English, but it does sound odd in American English. As Peter Shor says, for most other holidays, "Happy ---" is fine.
Best Answer
It's a misspelling[1] of Happy New Year's, i.e. Happy New Year's Day, where New Year's Day is January first. Or so I've always understood it, anyway.
[1] But see comments on this answer.