The logic behind this reasoning is the following: the Pater Noster is a paraphrase of Proverbs 3:19 which can be translated as following "God through (female) Sophia has stated the Earth and prepared the heaven in the mind"
ὁ θεὸς τῇ σοφίᾳ ἐθεμελίωσεν τὴν γῆν, ἡτοίμασεν δὲ οὐρανοὺς φρονήσει· LXX Septuagint Proverbs 3:19
And so Jesus is asking us to call for the Queen (Sofia) or Princess of God (ἡ βασιλεία) to come (ἐλθάτω) and finish what was prepared (ἡτοίμασεν), so that the wishes (θέλημά) of the God Father are born into existence (γενηθήτω). The language is clearly parallel to the Queen giving birth to the children of the Father.
ἐλθάτω ἡ βασιλεία σου, γενηθήτω τὸ θέλημά σου, ὡς ἐν οὐρανῷ καὶ ἐπὶ γῆς. Tischendorf 8th Edition 1872 Mt 6:10
Compare also to the Psalm 45(44) which is a song to the Beloved (son of God - Jesus) (ᾠδὴ ὑπὲρ τοῦ ἀγαπητοῦ) that talks about a Kings Daughter that will give birth to the sons that will become leaders around the earth.
1 Εἰς τὸ τέλος, ὑπὲρ τῶν ἀλλοιωθησομένων· τοῖς υἱοῖς Κόρε εἰς σύνεσιν· ᾠδὴ ὑπὲρ τοῦ ἀγαπητοῦ. ... 16 ἀντὶ τῶν πατέρων σου ἐγεννήθησάν σοι υἱοί· καταστήσεις αὐτοὺς ἄρχοντας ἐπὶ πᾶσαν τὴν γῆν. 17 μνησθήσονται τοῦ ὀνόματός σου ἐν πάσῃ γενεᾷ καὶ γενεᾷ· διὰ τοῦτο λαοὶ ἐξομολογήσονταὶ σοι εἰς τὸν αἰῶνα καὶ εἰς τὸν αἰῶνα τοῦ αἰῶνος· LXX Septuagint Psalm 45(44):16-17
The use of word ἡ βασιλεία is further seen in the declaration of what is happens from the times of John the Baptist who is from the earthly women counterpointing the heavenly (woman) Princess or Queen ἡ βασιλεία. And further the language of getting a wife through violence βιάζεται (rape) and following appropriation ἁρπάζουσιν is used. This may be also interpreted as a metaphor or as taking the kindom in a hurry or with effort but the other (sexual) meaning is certainly present as a linguistic reality.
ἀμὴν λέγω ὑμῖν, οὐκ ἐγήγερται ἐν γεννητοῖς γυναικῶν μείζων Ἰωάννου τοῦ βαπτιστοῦ· ὁ δὲ μικρότερος ἐν τῇ βασιλείᾳ τῶν οὐρανῶν μείζων αὐτοῦ ἐστιν. ἀπὸ δὲ τῶν ἡμερῶν Ἰωάννου τοῦ βαπτιστοῦ ἕως ἄρτι ἡ βασιλεία τῶν οὐρανῶν βιάζεται, καὶ βιασταὶ ἁρπάζουσιν αὐτήν. Tischendorf 8th Edition 1872 Mt 11:11-12
And at last even though all three forms of the word exist in Greek: ο βασίλειος masculine, η βασιλεία feminine, το βασίλειο neutrum, the feminine η βασιλεία is used in Pater Noster Mt6:10 (and Lk11:2) and Mt11:11-12.
[ῐ], ἡ, βασίλεια Pi.N.1.39: fem. of βασιλεύς:—A queen, princess, Od.4.770, A.Ag.84 (lyr.), Hdt.1.11, etc.; of goddesses, Κύπρις β. Emp.128.3, cf. Hymn.Is.I, etc.; β. θεά Ar.Pax974; β. γύναι A.Pers.623 (lyr.), E.El.988 (lyr.).
So apart from the translation tradition and maybe the absence of the genders in the target languages (like English) what speaks to the use of word Kingdom but not Princess (or Queen)? Should not we use the word Princess that actually reflects the meaning of the message?