You have some duplicate dates in your dataset, so that complicates matters a bit. The code below will match the two arrays.
%load data
s=load('bats_example.mat');
BATS_chla=s.BATS_chla;
BATS_nFLH=s.BATS_nFLH;
%keep only matching dates
date_chla=BATS_chla(1,:,1);
date_nFLH=BATS_nFLH(1,:,1);
a=ismember(date_nFLH,date_chla);
b=ismember(date_chla,date_nFLH);
BATS_nFLH=BATS_nFLH(:,a,:);
BATS_chla=BATS_chla(:,b,:);
%remove duplicate dates (keep the first)
date_chla=BATS_chla(1,:,1);
[~,idx]=unique(date_chla,'stable');
BATS_chla=BATS_chla(:,idx,:);
date_nFLH=BATS_nFLH(1,:,1);
[~,idx]=unique(date_nFLH,'stable');
BATS_nFLH=BATS_nFLH(:,idx,:);
%concatenate
output=cat(3,BATS_nFLH,BATS_chla);