This has nothing to do with the value of eps(0.0). x ≈ 0.0 will always give false for x ≠ 0, because ≈ tests only relative error. If x ≠ 0, then no significant digits of x match 0.0, so the result of x ≈ 0.0 is false.
In particular, x ≈ y tests norm(x-y) ≤ sqrt(eps(T))*max(norm(x),norm(y)) with T=typeof(real(x-y)), which tests whether about half of the significant digits match.
If you call isapprox(x,y) explicitly, you can pass an atol argument to instead pass an absolute tolerance, i.e. to test whether norm(x-y) ≤ atol. But since there is no reasonable way to pick a default value of atol (it is dimensionful, i.e. it depends on the scale of x and y), using ≈ sets atol to zero.
Note that 1.0 - 1e-10 ≈ 1.0 gives true, because more than half of the significant digits match 1.0.