The shirt is made of what fiber? Is it a natural fiber like cotton, linen, silk or wool, synthetic or a blend? What are the directions for care?
Is the whole shirt blue or just blue in spots? Did you soak the shirt in the detergent in an attempt to remove the blue stain or did the shirt become stained after soaking? What caused the stain that you treated by soaking? Did that stain come out?
Did you use fabric softener? It can leave an oily stain. Are you sure the Equate brand is chemically the same as the Dawn brand?
If you already dried the shirt in the dryer, the stain may be set.
I suggest calling Equate customer service. They will ask you some of these questions.
When home care products are colored, I expect they have dye in them. I've never heard it suggested that you soak an item in Dawn. The directions say rub the Dawn into the oily stain before washing it.
I use OxyClean for all my soaking. I leave the item in overnight. Recently, I soaked blood stained undershorts for 3 days. The stain was completely gone. I think I haven't been soaking blood stains long enough. That is the one stain I've had difficulty getting out of fabric.
I also apply liquid laundry detergent to stains before putting them in the washer. That removes recent stains most of the time.
ETA: I can't find anywhere on the Internet that Equate makes a dishwashing product. I learned that Walmart owns the Equate Brand and only sells pharmacy and beauty products. I don't know how you could contact the manufacturer. Or If the detergent you used is actually an Equate brand.