A resonant dipole antenna can be likened to a series RLC lumped constant circuit where the impedance seen across the two terminals contains zero reactance at resonance, and where operation at a frequency slightly above resonance results in an inductive reactance in the impedance, and a capacitive reactance in the impedance for frequencies slightly below resonance.
The conditions which determine resonance for a lumped constant RLC circuit which can be described using conventional circuit analysis are not the same as those seen for a system which is not very small compared to the wavelength of the frequency of operation and where wave theory mostly applies.
The formulas for inductive and capacitive reactance in a series RLC lumped constant circuit are shown below.
Xc = 1/2πfC
Xl = 2πfL
Where f = frequency, C = capacitance and L = inductance.
The formula for capacitance is C = e(A/d)
where e = dielectric permittivity, A = area of plates, d = distance between plates.
It's pretty obvious that a wire dipole antenna will have a very small amount of capacitance between the center feed points because the area of the plates, which is the surface area of the inner ends of the wire elements, is tiny. It is also obvious that this small amount of capacitance will not result in the same amount of reactance given by the self inductance of the wire elements, so these reactances will definitely not be equal and opposite at resonance.
In addition, see below the equation which describes the resonant frequency for a series RLC circuit.
fc = 1/2π√LC
A dipole antenna has multiple resonant frequencies, the above equation has only one solution for each combinations of L and C, and this equation cannot describe dipole resonance at harmonics of the fundamental resonant frequency.
See below equations using the Induced EMF method which are approximations which describe the impedance seen at the center feed points of a dipole.
where $a$ is the radius of the conductors, $k$ is the wave number 2πf/c, $η_0$ denotes the impedance of free space = 377Ω, and $γ_e$ is Euler's constant = 0.57721566
The terms in the equations cater for the length, diameter and self inductance of the dipole elements, as well as for the fact that each opposite point on the dipole elements is a different distance away from all other opposite points unlike the situation seen in a transmission line where the distance between opposite points is the same.
For a dipole antenna, the voltage of the standing wave on the antenna is at every point on the antenna about 90 deg out of phase with the current of the standing wave.
At resonance, the current of the standing wave is in phase with the voltage and current of the applied RF energy at the feed points.
At the feed points, the voltage of the applied RF adds vectorily to that of the voltage of the standing wave.
The reason a dipole has no reactance in the feed point impedance at resonance is because, at resonance, the electrical length of the dipole is such that the zero crossing point of the voltage of the standing wave (which is always out of phase with the applied RF), occurs exactly at the antenna center feed points and so contributes no reactance to the feed point impedance.
The electrical length is determined by the length and cross sectional area of the antenna elements, which includes self inductance and effects of the impedance of free space. This situation i believe is described in the equation above for feed point reactance.