= Solution
Combine the two equal-<mass> <Dirac fields> into $\Psi=(\psi^1,\psi^2)^T$. A constant <unitary matrix> $U$ acting on this flavour index commutes with the common <Dirac Hamiltonian>, so $\Psi\mapsto U\Psi$ gives the <unitary flavour symmetry of degenerate Dirac fields> $U(2)$. Its real <Lie algebra> has dimension four. A convenient Hermitian generator basis is $T^0=\mathbf1_2/2$, $T^a=\tau^a/2$, $a=1,2,3$, where $\tau^a$ are the <Pauli matrices>.
For each generator the <canonical anticommutation relations> give
$$
\rho_A=\Psi^\dagger T^A\Psi,\qquad
j_A^k=\bar\Psi\gamma^kT^A\Psi,\qquad
[\rho_A(\mathbf x),H]=-i\partial_k j_A^k(\mathbf x).
$$
The calculation is exactly the one in <3/b>, since $T^A$ commutes with both the <Dirac gamma matrices> and the common <fermion mass matrix>. Thus
$$
\boxed{\text{Four independent number-preserving vector currents:}\qquad
j_A^\mu=\bar\Psi\gamma^\mu T^A\Psi,\quad A=0,1,2,3.}
$$
Explicitly their densities are
$$
\rho_0=\tfrac12(\psi^{1\dagger}\psi^1+\psi^{2\dagger}\psi^2),\qquad
\rho_3=\tfrac12(\psi^{1\dagger}\psi^1-\psi^{2\dagger}\psi^2),
$$
$$
\rho_1=\tfrac12(\psi^{1\dagger}\psi^2+\psi^{2\dagger}\psi^1),\qquad
\rho_2=\tfrac12(-i\psi^{1\dagger}\psi^2+i\psi^{2\dagger}\psi^1).
$$
The two off-diagonal <conserved currents> are why counting only the individual <fermion numbers> misses two generators. The singlet generates $U(1)$; the other three generate <special unitary group> $SU(2)$. Unequal <masses> would break the off-diagonal <conservation laws>.
\b[Scope of the count.] Four is the count of currents of the displayed $\bar\psi^A\gamma^\mu T_{AB}\psi^B$ form. The completely free system also has an <enhanced internal symmetry of free Dirac fields> if mixing with <charge-conjugate fields> is admitted. For $m\ne0$, write its two <Dirac fields> as four equal-<mass> <Majorana spinors> $\chi^I$. Their <orthogonal group> $O(4)$ rotations have six continuous generators, with <conserved currents> proportional to $i\bar\chi^I\gamma^\mu\chi^J$ for $I<J$; the four $U(2)$ currents form a subgroup. This larger count is relevant if “similar” means \i[all] local internal currents of the free system, without preserving a chosen <fermion number>. If $m=0$, the usual number-preserving <chiral flavour symmetry of massless Dirac fields> is $U(2)_L\times U(2)_R$ with eight currents; allowing <charge conjugation> mixing enlarges it to $U(4)$ with sixteen. These counts concern internal currents, excluding spacetime <Noether currents>.
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