Don't stop, running is ace.
When you start doing races you will see (as I'm sure you can guess) that there is always someone faster than you - and always someone slower. You can only meaningfully compete with yourself, race to race.
He can only compete with you if you tell him your times, so don't. I understand you my have wanted to be able to talk to him about it and gain support and now you feel you can't but that is all you've lost.
I think you should tell him that you don't really appreciate your hobby being turned into a competition and would rather not hear about his progress. He'll probably say he's not competing (of course he'd naturally be better at it) he's offering positive encouragement. All he needs to know is that you find it off-putting and don't want to hear about it. He only needs to believe you, not to understand your position.
Could you join a running club? Not necessarily a women only one, club members are more experienced and sensible, they know they're competing against themselves, supporting each other just by running together with same-speed people and that anyone can get injured and suffer setbacks. I've found them a very benign, positive influence.
I found I could increase steady distance running alone but a club was fantastic for increasing speed, by doing sessions with the short bursts at higher speed, or uphill, that you need to make muscles stronger, thus get faster at longer distances. I find that sort of training much easier and more fun to do with other people.
You'd need to be able to run about five miles regularly to join a club, unless they do a specific 'new runners' session but you don't need to be fast. Best if it's a good size club so there'll be plenty of people around your speed.
Otherwise or anyway, just keep going, start doing some races, park runs, 10kms, they're great training and give you your own benchmarks. If your half is only a month away though, don't go crazy and remember to taper - reach peak training in the 3-1 weeks before then tail off, so your muscles have time to strengthen and recover before the race.
After that, I find 'non-runners' often focus on training for marathons. It's the most prevalent idea of 'what is it to achieve something at running'. It's really just another distance, that may or may not suit you. Training for a marathon takes much, much more time than for a half. I'd rather improve my 10k and half times.
Finally, if you're a lot lighter than him, you'll perform relatively better over longer distances. So he'd out-sprint you, be better over 1 or 3 miles. But, if you're about the same at 10k, you'd be faster at half marathon. Maybe he's faster at all distances but the relative difference could be there. Once you get really fit and into it, you'll be past caring though. Have fun 