It does sound like ARFID.
Did she have reflux or silent reflux as a baby?
She could have physical issues with swallowing. Or she could have ARFID relating to neurodiversity - no signal from brain to tell her she is hungry, profound dislike to all sorts of flavours and textures.
She's old enough now to explain to her what food is for. Explain that food gives you energy to play, grow tall and strong, make your brain clever etc and that hidden inside food are little units of energy called calories and she needs enough of them each day.
DS2 is autistic and had severe ARFID. He was just like this. I honestly don't know how he stayed alive. Two bites. no more, ever. I learned to pack as many calories into those two bites as it is humanly possible to do. (Tiny postage stamp sized sandwiches with full fat cream cheese, peanut butter and jam half the time.)
But if she has eczema, she may have a lot of food sensitivities. Have you had her tested for nuts and peanuts? If she is okay with them, can you try sneaking nut butter into her, as it is pretty much the most high calorie food you can eat in tiny quantities.
Can/do you toss pasta, carrots, chips etc in a bit of extra butter and oil? Not enough to make her taste it or make them greasy but even a tiny shaving of butter is calorie dense.
Have you tried letting her eat UPFs? Counter-intuitive but rules for ARFID children are almost the opposite of rules for feeding other children. DS would sometimes eat lots of cheesy wotsits at kids' parties. Better to have calories from any source at all than none.
Do you add Abidec vitamins drops to her milk? (Highly recommend you start this if you haven't yet)
Keep introducing foods and explaining the powers they will give her: salmon makes you clever, chicken makes you strong, pasta and chips help you run around, carrots and other fruit and veg help your body fight illness etc. If she is neurodiverse, it takes (ime) about 2000 introductions of a food before a child will trust and try it. Neurotypical children are familiarised with new foods after twenty introductions to it, but for autistic children multiply that by a hundred. Don't give up. I was in despair for years about DS's eating. Because I had no idea he was autistic, I didn't realise he was likely to have a very restricted taste in foods so I kept on offering them without pressure, for years and years and years. He's now an adult and eats as wide a variety of foods as any other adult. In retrospect, I'm glad I kept trying though it was a massive source of stress at the time.