Yes it can. :) You can write "iloista syntymäpäivää" (happy birthday) on a card, and even say it, but in my experience "paljon onnea" is much more common. In the Finnish version of the "happy birthday to you" song people sing "paljon onnea vaan".
'Onni' can mean happiness or luck depending on the context. "Paljon onnea" is usually used to wish someone a lot of happiness in life, so it doesn't really work for wishing luck - for that usage you can just say "onnea" (or use colloquial sayings like "lykkyä tykö" etc.)
There should be "congratulations" (for birthdays or achievements) used in these translations rather than "good luck" (before games or performances) or "best wishes" (sending regards via letters/mail) most of the time.
Would paljon onnea also translate as, "many congratulations" and/or "much good luck!"? I guess "much good luck" means something very similar to "best wishes" anyway. Are there other expressions for those? Seems like Duo should take those if they do indeed work.