Judaism often relates to descriptions such as son and father as metaphorical, regarding the relationship between God and the Jewish people.
The daat mikra Tanach explains the psalm to be divided into 4 parts with different speakers, verses 1-3 being the non Jewish nations, 4-6 being God to the nations 7-9 being the king relating God's words to him and 10-12 being the king talking to the nations.
Traditionally (based on chazal) the whole psalm is a description of conversations and rebuke to the non Jews between God the nations and the king mashiach. Specifically, in verse 6 God refers to his prince, or annointed one, the mashiach.
However, there are also other understandings (e.g. Rashi) that read the psalm as a description of one of David's wars, to remind the Jews about to fight another nation of God's protection for them.
Specifically regarding verse 7, in the messianic understanding, as I hinted to earlier, daat mikra explains the references to son as follows (description/translation of the commentary mine):
...God said to me, you are to me as a son and from now on to the future I will be considered your father (rather than your actual father).
In other words, the Jewish king, or mashiach, is being described as a metaphorical son; this is a device used throughout Tanach.
Additionally, the translation you have should not read "Today I have given birth to you" rather it's a (metaphorical) description of the future; "I have given birth" (past tense) combined with"today" means from now on.