Are they both correct and mean the same?

the year-end and new year holiday

the year-end and new year holidays

Do you mean the two days - New Year's Eve and New Year's Day?
If it's one block of time, then I'd probably keep it singular, but either is okay.
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Hi, Annvan

I mean a week-long holiday(s?) which usually runs from Dec 28 through Jan 4 in my country.
 BarbaraPA's reply was promoted to an answer.
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Thanks a million, GG.