Blender results in that state, because there is not UVmap delivered with obj.
On other hand, importing your OBJ to Meshlab doesn't use your texture at all. I would say it uses Vertex Color info ... the weird thing is imported OBJ to Blender doesn't have any Vertex Color either, even when I export obj from meshlab with vertex color ... hm.
Before than someone smarter will answer this "mystery" ... You have two options - open your OBJ file in Meshlab and Export as PLY format. There I have no issue to use the Vertex Color.

If you want to use Image Texture, you would have to create UV map in Meshlab first and than transfer Vertex Color to Texture. For more info see https://stackoverflow.com/a/51582851/10148444 ... this worked for me too.