hello,
It is a quite interesting question and I think that, with some remarks, there is no reason why an isotropic antenna cannot be made.
remark 1: The IEEE says that an isotropic radiator must be lossless. this is impossible
remark 2: One polarisation must be selected (it is not possible to make an antenna sensitive to all polarisations -linear, horizontal, circular Rh +Lh-.
remark 3: don't think of a single antenna (it should lead to a point, which is not possible), think of a 3D array
therefore the design could be a 3D array, sensitive to one polarization and with some loss, but with a nearly-spherical radiation pattern.
if we consider a cube with one antenna on each facet (I think on patch antennas with directivity of about 6dB), and with a power divider 1:6 feeding all the units, the pattern should be nearly-spherical, sensitive to a field component (theta for instance). The pattern will have ripples, due to the interaction between closer elements patterns, but even with that the pattern should resemble a sphere.
regards