A multitape Turing machine is defined to have input only appear on one tape, with the rest of the tapes blank.
Are there any formulations of a Turing machine that allow other tapes to be not blank? Why does the condition that the input only appears on one tape exist?
It is just simpler to describe the situation with a single input tape -- in particular, if you want to compare the power or efficiency of machines with differing number of tapes, it is convenient that the format of the inputs to all the machines are the same.
But it is not a particularly deep choice -- if you have input on one tape and want it spread across several, or have input on multiple tapes and want it combined into one, it is generally trivial to specify a machine that starts by rearranging the inputs as you wish.