I read the Schmidt basis is unique "up to a phase" or as stated here in answer given by Norbert Schuch "modulo degeneracies".
If I choose a Bell state
$|\psi\rangle = \frac{1}{\sqrt{2}}\left(|0\rangle\otimes|0\rangle + |1\rangle\otimes|1\rangle \right)$
and then define a second basis on each of the two subspaces by
$ |0\rangle \equiv \frac{1}{\sqrt{2}}(|\tilde{0}\rangle +|\tilde{1}\rangle) \quad \quad |1\rangle \equiv \frac{1}{\sqrt{2}}(|\tilde{0}\rangle -|\tilde{1}\rangle) $
then $|\psi\rangle$ becomes
$|\psi\rangle = \frac{1}{\sqrt{2}}\left(\frac{1}{\sqrt{2}}(|\tilde{0}\rangle +|\tilde{1}\rangle) \otimes\frac{1}{\sqrt{2}}(|\tilde{0}\rangle +|\tilde{1}\rangle) + \frac{1}{\sqrt{2}}(|\tilde{0}\rangle -|\tilde{1}\rangle) \otimes\frac{1}{\sqrt{2}}(|\tilde{0}\rangle -|\tilde{1}\rangle) \right) =$
$\frac{1}{2 \sqrt{2}}( |\tilde{0}\rangle\otimes|\tilde{0}\rangle + |\tilde{0}\rangle\otimes|\tilde{1}\rangle + |\tilde{1}\rangle\otimes|\tilde{0}\rangle + |\tilde{1}\rangle\otimes|\tilde{1}\rangle + |\tilde{0}\rangle\otimes|\tilde{0}\rangle - |\tilde{0}\rangle\otimes|\tilde{1}\rangle - |\tilde{1}\rangle\otimes|\tilde{0}\rangle + |\tilde{1}\rangle\otimes|\tilde{1}\rangle )$
finally
$|\psi\rangle = \frac{1}{\sqrt{2}}\left(|\tilde{0}\rangle\otimes|\tilde{0}\rangle + |\tilde{1}\rangle\otimes|\tilde{1}\rangle \right)$
which again is a Schmidt decomposition with the same coefficients as above but with different bases. What is my mistake?