I'm 31.
My partner n I were making love last night when suddenly it felt like his penis pushed through something and he fell deeper inside me. We both felt it and it bled a little.

I remember the first time I had sex. I used to be married and married my first love. It didn't hurt and I didn't even feel like we broke/tore through anything or even went through anything. But there was the slightest amount of blood.

Last night... I felt a slight discomfort and his penis pushed through something... And there was blood, and it was more than my first time.

He said it felt like "I was pushing ok and then something gave way and I could go deeper, and it was slightly warm once I got through"

I had a slight discomfort inside, but it didn't last long.

I tried to Google what this could be but from what I read, it seems like I only just broke my virginity/hymen?
I read what many girls describe what it's like to break their hymen... And I've read what it feels like for guys to break through a girls hymen... And it's exactly what we feel we felt last night.
It seems that I may have had a rigid/tight/elastic hymen...

Could it be possible mine never really broke/tore?

Could it be possible that my hymen is a bit deeper?

Yes to all the above. Check with your OB/Gyn

Only your doctor can tell you that.