The frequency response function (FRF) fails to detect the antiresonance of a system
I agree with A_A that the chirp is too fast.

I adjusted a few things in the code


increase time to $600$ (seconds)
start chirp at $0.5\textrm{ Hz}$
end chirp at $2\textrm{ Hz}$
removed zero padding (not required here)


And the result is...



The full code is 

% Identification of the Frequency Response of a Transfer Function with
% Resonance and Antiresonance peaks
%% Cleaning the house
clc;
clear;
close all;
%% Definition of the System 
wn_z1 = 10*2*pi;           % Anti-Resonance Natural frequency 10 Hz
wn_p1 = 80*2*pi;           % Resonance Natural frequency 80 Hz
chi_z = 0;                 % Damping factor of the zeros 
chi_p = 0;                % Damping factor of the poles
s     = tf('s');
sys_1 = (s^2+2*chi_z*wn_z1*s+ wn_z1^2)/(s^2+2*chi_p*wn_p1*s+wn_p1^2);             
zpk(sys_1)
%% Definition of the Chirp Input
h                    = 0.001;
time                 = 0:0.001:600;
freq_ini             = 0.5*2*pi; % Initial frequency considered (0.1 Hz)
freq_final           = 2*2*pi; % Final frequency considered (240 Hz)
chirp_input          = 1.5*chirp(time,freq_ini,time(end),freq_final,'logarithmic');

frequency_evolution  = logspace(log10(freq_ini),log10(freq_final),length(time));
figure(1);
subplot(2,1,1);plot(time,chirp_input); xlabel('time (s)'); ylabel('Chirp    amplitude')
subplot(2,1,2);plot(time,frequency_evolution/(2*pi)); xlabel('time (s)'); ylabel('Frequency (Hz)');
%% Execution of the simulation
[sys_output,~]       = lsim(sys_1,chirp_input,time);
%% Computation of the FRF
fft_input   = fft(chirp_input);                                           %% Considering zero padding
fft_output  = fft(sys_output );                                           %% Considering zero padding
fft_input   = fft_input(1:length(fft_input)/2);
fft_output  = fft_output(1:length(fft_output)/2);
fft_output  = fft_output(:);
fft_input   = fft_input(:);
FRF              = abs(fft_output)./abs(fft_input);
frequency_vector = linspace(0,1/(2*h),length(FRF));

%% Comparing the Bode Plot (Matlab) with the FRF (own implementation)
opts = bodeoptions('cstprefs');
opts.FreqUnits = 'Hz';
close all;
bodemag(sys_1,opts,'r'); hold on;
semilogx(frequency_vector, 20*log10(FRF ),'b')
xlim([0.1,frequency_vector(end)])
legend('Matlab Bode','FRF Identification')