The hymn “O Come, All Ye Faithful” was originally in Latin, and even today is still often sung that way under the title “Adeste Fideles”.
We are not certain who wrote its original tune or lyrics, although this was probably in the 1700s or perhaps the 1600s. We do know, however, that it was translated into English in 1841 by a Catholic priest.
Venite adoremus Dominum.
Has Dominum (meaning Lord) in the accusative; it would have been Domine in the vocative and Dominus in the nominative. Both verbs can be read as plural imperatives, the first in the second person plural, the second in the first person plural. Hence the familiar translation:
O come let us adore Him, Christ the Lord.
The verse you mention:
Yea, Lord, we greet thee;
Born this happy morning;
Jesu, to thee be all glory given!
Is a translation of the Latin:
Ergo qui natus die hodierna.
Jesu, tibi sit gloria,
Latin used Iesus/Jesus in the nominative, Iesum/Jesum in the accusative, and yea Iesu/Jesu for everything else, notably including the vocative. This is highly irregular for Latin declensions, where it fits nicely in no declension in particular, although it tends to be attributed to the fourth (not the second). It really doesn’t matter, though, what declension it’s considered so long as all the forms are known — and the gender, since that’s needed for adjectival concordance.
It has all those funny forms in Latin because it was borrowed from the Greek, where if you transliterate the declined Greek forms into Latin, you’ll see why it’s done that way,
Those funny Latin forms were even copied into the German declensions, where again it sticks out like a sore thumb from the normal scheme of things.
So in all these cases, it’s because scholars familiar with more than one language were consciously copying the earlier declensions into the later, translated language no matter whether it fit that language or not.
It’s like how we still sometimes use a few Latin plurals in English, or the rex–regina distinction. Here, the Catholic priest who translated the Latin into English merely left the inflected form intact. But ultimately he has the Greeks to thank for the strange declension.
Lastly, I have sometimes seen Jesus used here in some modern versions of the song, not the original Jesu. I cannot say whether this is an intentional updating or not, however; the rest is unchanged.