This is, of course, an opinion from a very novice gardener here, but if you were to ask me I'd say that this is definitely ordinary... well, I'm not sure what it's specifically called, but ordinary crab grass? At any rate, it's found everywhere and often takes over in MY garden, so you'll want to probably uproot it. Careful of those darn seeds spreading. If you've got any animal friends (like, guinea pigs, cows, horses, sheep) who happen to live nearby, they'd love it. lol But again... I'm fairly new at this, so don't take my word for it.

The only ferts my grasses get is an occasional bit of compost tea, very dilute. Apart from that, they get nutrients from decomposing mulch and that's it. Ornamental grasses really don't benefit from overly fertile soil, and can get too floppy if there's too much nutrient. I remember reading, on this forum, someone's experience with lawn ferts and A. donax, and I believe it was a positive one. I've never grown it, so I can't speak to it's need for ferts. I wouldn't put any Weed-n-Feed type stuff anywhere near any ornamental grasses, but I don't use any of it anyway. I'm trying to get rid of lawn, not encourage it.

I can't tell from the picture. Could you post one that shows a closer view of the grass? My best guess would be annual bluegrass. If that is what this is in your yard, then it will die out as it heats up. I have heard that products like Image work, but don't care to use them on my own property.

It looks like Johnson grass. If it is, be careful removing it, the blades (sides) are razor sharp. I've been sliced by them many times. Dig it out, and wear heavy gloves.