well the one you showed wasn't embroidered, it was woven that way. it is also stitched like a tube to allow for a seamless design front and back. pretty much all scarves you get will be like that. embroidery will work, just make sure they also do the seamstress work to make them and quote in full. expect the embroidered to be hella expensive.
Now bear with me on this next part ... it's been a decade since I've had to this next one done so I can't remember the name. It's the printing process normally done on photos where the image is printed from flat sheet materials that get atomized into a hot gas to empregnate a material. Well anyways, you normally find it in the sign industry used with fire retardant materials like scrim and polyester banners. I know they also use it for jersey material and thicker softer fabrics ... might need to get the material printed in multiples and then do the seamstress work yourself ... but it will be a softer graphic for around the neck ... still would sew it like a tube too.
And of course if you can find a wholesale solvent printer who can print on softer cotton fabrics ... there is always that route too.