The translations you cite in (the original version of) your question are not Jewish translations, despite the misleading name of one of them. Let's look at this verse in more reliable renderings.
First, the Hebrew:
כִּסְאֲךָ אֱלֹהִים עוֹלָם וָעֶד שֵׁבֶט מִישֹׁר שֵׁבֶט מַלְכוּתֶךָ:
JPS translates it this way:
Thy throne given of God is for ever and ever; a sceptre of equity is the sceptre of thy kingdom.
Sefaria has (from JPS 1985):
Your divine throne is everlasting; your royal scepter is a scepter of equity.
These two translations treat אֱלֹהִים as an adjective modifying כִּסְאֲךָ (your throne). In Biblical Hebrew, adjectives come after the nouns they modify. According to this interpretation, God (אֱלֹהִים is one of God's names) is not being addressed but, rather, cited.
Rashi interprets it differently:
Your throne, O judge, [will exist] forever and ever; the scepter of equity is the scepter of your kingdom.
Your throne, O judge: Your throne, O prince and judge, shall exist forever and ever, as the matter that is stated (Exod. 7:1): “I have made you a judge אלהים) (נתתיך over Pharaoh.” And why? Because “a scepter of equity is the scepter of your kingdom,” that your judgments are true, and you are fit to govern.
Rashi is interpreting אֱלֹהִים as "judge" based on another verse that, according to Onkelos and Tanchuma, uses the word that way. In that verse (Sh'mot 7:1), the judge (Moshe) has a divine mandate, so this isn't as big a difference as it first appears.
So, the psalm does not address anybody as God. This verse addresses the same "you" as the rest of the psalm -- "I speak my poem to a king" (v2). From context this is an earthy king, not the King.