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For any lattice a good choice of a unit cell is the one that is the most symmetric.

Now if every lattice made by a body centered tetragonal unit cell can be made by a face centered cubic unit cell ,which is more symmetric ,then why keep the body centered tetragonal unit cell as a class of Bravais lattice ? enter image description here

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    $\begingroup$ Because not every body centered tetragonal cell is equivalent to a face centered cubic cell. $\endgroup$
    – Ivan Neretin
    Commented Nov 29, 2020 at 15:13

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Body-centered tetragonal is face-centered cubic only if $c/a=\sqrt2$. If you try your transformation with a $c/a$ value greater/less than $\sqrt2$, your "cube" will have lateral edges that are longer/shorter (resp.) than the basal edge and so really remains tetragonal.

You can also read this the other way. If you are presented with a body-centered tetragonal lattice and $c/a$ seems to be close to $\sqrt2$, maybe it's really fcc.

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