It is a shortcoming of the software, which only deals with the generic case.
DSolve[{y'[x] == y[x]/ z[x],
z'[x] == z[x]/ y[x],
y[0] == 1, z[0] == 1},
{y, z}, x]
gives empty solution set, while this
DSolve[{y'[x] == y[x]/ z[x],
z'[x] == z[x]/ y[x],
y[1] == 1, z[1] == 1},
{y, z}, x]
gives a funny Indeterminate
output, when it should give y[x] == z[x] == x
.
The option IncludeSingularSolutions
does not help in this case.
The linear solutions can be recovered for example this way:
eqs = {y'[x] == y[x]/ z[x],
z'[x] == z[x]/ y[x]};
linearSolutions = {y -> Function[x, a x + b],
z -> Function[x, c x + d]};
linearSolutions /. SolveAlways[eqs /.
linearSolutions,
x]