Mongolians and Native Americans, to my knowledge, share a common ancestral home in the steppes of Asia. Both Mongolians and plains dwelling Native Americans have temporary dwellings in the forms of yurts and tipis, respectively. Did these designs evolve independently of each other or was one the precursor of the other? Or perhaps is there an older common ancestor of the two?
Native Americans were migrated from today's Siberia and they are sharing some genetic and cultural relationships with people from these area. However, their similarity is together with Siberian Turks rather than Mongolians. Nearly all of Native Americans carrying y-dna haplogroup Q and R and it is mostly seen together in Turkic peoples especially Tuvans. They probably shared genetic and cultural pool but Native Americans migrated too much before Turks or any nations emerged. So its not meaningful to say commoon ancestor because most of the nations sharing common ancestors by that time.