I think you want to do something like Generate Near Table or just Near which will allow you to join attributes between the points for each census tract and the near table to find the distance to the city centres.
If you use closest only on the near table you will only get the closest feature, Near will always return just the nearest feature and distance from one feature class to another, in addition it should add Near_FID to the input points so you can identify which one by joining Near_FID to the FID on the city centeres.
To find the distance between census tracts and the matching code number:
Generate near table using tracts as input and city centres as near features with all distances and a very large or not entered tolerance; the documentation says If no value is specified, all near features will be candidates which should mean that every census tract to every city will be represented in the table should no tolerance be entred, if it is not the case then use a very large tolerance.
From here there are many ways to trim that down, but this is the method I use as it's less confusing (for me anyway):
Add two fields to the near table, let's call them TractCode and CityCode. Load up the table in ArcMap (you will need to use add data as tables don't want to drag 'n drop), add the census tracts and city centres as well.
Open the attribute table for the near table and Add a join from the near table to the city centres using NEAR_FID to FID/OBJECTID (depending on data type). Using field calculator calculate the value of the CityCode in the near table from the code field in the city centres data. Now remove that join and add another one to the census tracts and calculate the TractCode from the census tracts and then remove that join too.
Now your near table has the codes from the census tracts and city centres. Select by attributes on the near table where CityCode = TractCode which will select the rows where the code is the same on city centre and census tract, export that to a new table for simplicity.
This newly exported table will contain the distances between census tracts and city centres where the code matches.