Axial or planar chirality in spiro[3.3]hepta-1,5-diene
Useful Definitions (from the IUPAC Gold Book)


chirality plane - "A planar unit connected to an adjacent part of the structure by a bond which results in restricted torsion so that the plane cannot lie in a symmetry plane. For example with (E)-cyclooctene the chiral plane includes the double bond carbon atoms and all four atoms attached to the double bond; with a monosubstituted paracyclophane the chiral plane includes the monosubstituted benzene ring with its three hydrogen atoms and the three other atoms linked to the ring (i.e. from the substituent and the two chains linking the two benzene rings)."
planar chirality - chirality resulting from a chirality plane.
chirality axis - "An axis about which a set of ligands is held so that it results in a spatial arrangement which is not superposable on its mirror image. For example with an allene abC=C=Ccd the chiral axis is defined by the C=C=C bonds; and with an ortho-substituted biphenyl the atoms C-1, C-1', C-4 and C-4' lie on the chiral axis."
axial chirality - chirality resulting from a chirality axis


Interpretation


According to the above definition, a chirality plane must involve restricted torsion (restricted rotation) about a bond.
A molecule can contain both a chirality plane and a chirality axis; an example would be an ortho-substituted biphenyl where the restricted rotation is responsible for chirality (atropisomers).


Conclusion


  Axial or planar chirality in spiro[3.3]hepta-1,5-diene


There is no restricted rotation that brings about chirality in this molecule.  Unlike trans-cyclooctene or a biphenyl, this molecule cannot be made to lie in a plane by torsional motion (at least not as long as we maintain the tetrahedral nature about the spiro carbon). Therefore, this molecule does not possess a chirality plane.

Rather, this molecule is analogous to the allenes. It does have a chirality axis passing through the $\ce{CH}$ group in each ring furthest from the spiro carbon along with the spiro carbon. The molecule can be said to possess axial chirality.