Revelation 3:9 is Jesus speaking to the church in Philadelphia, a small congregation being harassed by a local group that claimed spiritual authority they didn’t have. The promise is that God will vindicate His faithful people, and the ones opposing them will recognize God’s love was with that church.
You’re right to be cautious about how this verse has been twisted. “Synagogue of Satan” refers to a specific group in that city who were opposing Christians and likely getting them in trouble with the authorities (the church had lost its legal protection, so that was a real threat). Jesus is saying their claim to represent God rings hollow because of how they’re treating His people. Reading the verse as a blanket statement about Jewish people gets the context backward.
Verse 8 says this little church had “little strength” and Jesus promises they’ll know He loved them.