כַּאֲשֶׁ֨ר הָיָ֤ה יְהוָה֙ עִם־אֲדֹנִ֣י הַמֶּ֔לֶךְ כֵּ֖ן יהי [יִֽהְיֶ֣ה] עִם־שְׁלֹמֹ֑ה וִֽיגַדֵּל֙ אֶת־כִּסְא֔וֹ מִ֨כִּסֵּ֔א אֲדֹנִ֖י הַמֶּ֥לֶךְ דָּוִֽד׃
So wie Gott mit meinem Herrn, dem König, war, so sei er mit Salomo, und erhöhe seinen Thron noch über den Thron meines Herrn, des Königs Davids.
Rabbeinu Bahya
רק הכסא, “only by the throne, etc.” Pharaoh meant by this that he would be the titular head of state, i.e. that the people would call him “King.” We find in Kings I 1,37 ויגדל את כסאו, “may he make his (Solomon’s) ‘throne’ greater than that of his father.” This too is a reference to Solomon’s reputation as a great king being established.
Rashi
כסא THRONE is a metaphorical term for royal rank, like (1 Kings 1:37) “And he has made his throne (כסאו) greater than the throne of my lord king”.