Two modals one right after one - are correct? I heard yesterday someone who seems to be a native English speaker who said "I would wish..." (I didn't listened further) and It was weird to me since I learnt that modals considered a sort of auxiliary verb, and since it doesn't make sense to say "I will will" or "I was was" the same it will not be correct to say "I would wish". So if it's not correct I can understand why, but if it's correct I don't understand why it's like that. Is there a way in which it's correct to say "I would wish..."?
Wish is not a modal verb, so can follow would or any other modal verb.
It is perhaps slightly awkward to say "I would wish ..." by itself, but if I asked you "If you could wish for anything in the world, what would you wish for?", you could reply "I would wish for ...".
You have misunderstood the website you linked to, which is about the modal verb may, not about the main verb wish. It uses the modal verb may to express a request, or permission, or prohibition, or possibility or a wish. The example is "May the New Year bring you joy".