Looks more like Mondo Grass, Ophiopogon japonicus. Where are you located? That matters for plant ID.

Im in NC, I got it from nursery about 14-15 years ago. Since it was long time ago, I dont remember what name was.

What part of NC? Mondo Grass needs zone 7 or higher.

Eastern NC, about 50 miles east from Raleigh, according to this map it should be 8 - (yellow part) https://gilmour.com/north-carolina-planting-zones

I googled pictures - looks like mondo grass have blue berries, but my grass have black.

I changed my mind despite your zone 8. I did some googling too. Probably the Liriope spicata. Maybe someone else will come along and agree - or not. You've had if for 15 years now, seen how it spreads and been fine with it. Both the Mondo and L. spicata wants to be a groundcover.

The quality of the invasion can vary greatly with site conditions, time and maintenance habits.

I never thought it spreads badly, actually stays in that area without me doing anything special. I did spread from little plants I bought to around 4 sq ft area, but usually all I do about maintenance - is trimming in spring time to remove old leaves. This is the first time I pruned it near border, I never did it before in all those years. I read that border for it should go like 10"+ deep in ground, but that little border I have maybe just 2-3" deep. Not sure if roundup or lawn moving helps, but those things I do regardless if it there or not.

People on google telling horror stories how they have hard time containing it, also berries should be dark blue, not black, and leaves 1/4 wide - this why I wonder what type I have, and whatever it will behave more aggressive in different spot. To me it looks like cross between two :)