No, inflammation won't cause a false positive, but do you have the actual test results?
Most people with hsv1 never get a cold sore - something like 90% of them don't. It's not surprising that you wouldn't either. It's quite possible it's something you got as a child from someone kissing you, or sharing a toy that you put in your mouth, or even sharing a drink. I wouldn't look at this as an STD, or even assume it's a genital infection unless you have an outbreak genitally.
Your partner should test, though. It's quite possible he already has it, too.