My name is how others identify me, how I introduce and represent myself; it symbolizes my character and nature. If someone disrespects my name, or leverages my reputation without my consent to further themselves, I take it personally.
Taking someone’s name in vain is to employ it in a light, casual or inappropriate manner, for any other purpose than to refer to them and honor them. Doing so treats their name as if it’s void of proper meaning and significance, empty.
So, God is grieved as we speak His name in vain (Ex 20:7), lightly, in an empty manner, when we aren’t referring to Him. To use His name as an interjection or expletive, as an expression of intensity or emphasis (e.g. J!, JC!, OMG! or GD!), is ultimately to disregard and despise Him. This is done so casually today, even by those who profess to believe in Him, we might have become desensitized, callous to it. God never gets used to it.
I never hear any false deity’s name ever taken in vain, only the true. What would be the point of insulting a non-entity? There’s only substance in trampling deity underfoot if He’s real, if He’s angered by the disrespect.
Should it grieve us, as it does Him? I think so.
But there’s a more direct application of this concept, which is evidently more aligned with the intent of this 3rd command. The word take in the Hebrew is נָשָׂא, nasah, to lift, carry, take up. It describes how the high priest bears the names of the 12 tribes of Israel upon his heart (Ex 28:29), associating himself with them and representing them before God in the Holy of Holies.
To take up, or associate ourselves with God’s name inappropriately, claiming we’re His representatives in order to elevate and enrich ourselves, when He is not aligned with us, when our hearts are not sincerely aligned with and subject to Him, this is potentially much worse than merely speaking His name inappropriately. In this way, we falsely represent Him to others, who then associate God Himself with our sins and indiscretions and blaspheme Him because of us. (Ro 2:24)