Albert Barnes Commentary Psalms 81:9

"There shall no strange god be in thee; Neither shalt thou worship any foreign god." — Psalms 81:9 (ASV)

There shall no strange god be in thee - Worshipped by you; or recognized and regarded as a god. This was a condition of His favor and friendship. Compare Deuteronomy 32:12; Isaiah 43:12. The word here rendered “strange” - זר zār - has reference to one of a foreign nation; and the meaning is, that they were not to worship or adore the gods that were worshipped by foreigners. This was a fundamental law of the Hebrew commonwealth.

Neither shalt thou worship any strange god - The Hebrew word here is different - נכר nêkâr - but means substantially the same thing. The allusion is to gods worshipped by foreign nations.