Onnea means both congratulations and (good) luck. It was confusingly introduced mainly with good luck even though congratulations is a lot more common. Paljon onnea literally means lots of luck or lots of congratulations. I wouldn't mix it with wishes since we have other more accurate translations for that.
Thanks @Puikelsson I searched Wiktionary again for "onni" and it does say "happiness" or "luck", whereby "onnea" is in the partitive case. https://en.m.wiktionary.org/wiki/onni#Finnish
When I meant "wish", I did not mean "desire" but a sense of "hope that one has luck". In English the idiom "best wishes" is probably just short for "best wishes of luck to you" or something like that.
Yes, I'm a native. I don't have a very strong dialect since I grew up influenced by two and needed to be easily understood and accepted my both. I use puhekieli with sä and mä and write kirjakieli on forums etc. Lappi dialect would be very different with all the added extra H's :)
I'd say that this a weird translation. "Paljon" simply means a large quantity—words like lot, much and many comes to mind; so "much good luck" is a closer translation, but of course nobody would say this in English. This shows why direct translation isn't a very effective way to learn a new language.