Mary is made by Luke to be Elizebeth's 'relative', Elizebeth (JtB mom) being called a daughter of Aaron hence of the tribe of Levi. The WTS and others then proceed to insist this legitimizes Jesus as a Levite and a Judahite and therefore qualified to be priest and King. To keep my comment short I'll just cut to he chase; the Epistle to the Hebrews (Hebrews) chapter 7 specifically argues against Jesus having been of the tribe of Levi.
11 If indeed, then, perfection were through the Levitical priesthood -- for the people under it had received law -- what further need, according to the order of Melchisedek, for another priest to arise, and not to be called according to the order of Aaron?
12 for the priesthood being changed, of necessity also, of the law a change doth come,
13 for he of whom these things are said in another tribe hath had part, of whom no one gave attendance at the altar,
14 for [it is] evident that out of Judah hath arisen our Lord, in regard to which tribe Moses spake nothing concerning priesthood.
15 And it is yet more abundantly most evident, if according to the similitude of Melchisedek there doth arise another priest.......
The force of the author's creative argument for the necessity of a priest in the order of Melchizedek hinges upon Jesus being understood as not of the tribe of Levi.