Meaning of "auto" in trans autophosphorylation? Why this process called "auto"? Is it because each tyrosine kinase receptor subunit of the RTK dimer has the ability to phosphorylate tyrosine or other amino acid residue present in other subunit of the RTK dimer's cytoplasmic tail? Is it because no new set of kinase enzyme is needed to phosphorylate them?

Receptor tyrosine kinases are proteins which float around in the cellular membrane. Upon binding of their ligand, the dimerize (two units form a dimer). This releases the kinase domains and the proteins start phosphorylating themselves over cross (meaning part a phosphylates part b and vice versa). This looks like in this figure below (from here, it is an interesting article):

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Besides ATP this process needs no further co-factors so the "auto" refers to the ability of the RTK to phosphorylate itself upon activation. See more about autophosphorylation here.