Why can't listing agents just add a note that MLS Sq ft doesn't match with City's Sq ft but he/she has personally measured it. It will save a lot of work for quite a few buyers and their agents.
Most real estate professionals attempt to provide accurate information on a property. You are correct to doubt any information because at the bottom of each page there is a disclaimer indicating all information may not be accurate.
You the buyer are responsible for performing your own Due Diligence on any property. You and your Buyer's Agent have the right to measure the property yourself and make appropriate adjustments.
Your lender (assuming not a cash purchase) usually requires an appraisal and they may provide another set of values for sqft and price per sqft.
I suggest discussing this with your real estate agent further and taking action as required based upon facts not included in your question.
I looked back in the listing history of the home and the home has been listed a few times by different agents with 1540 citing the source as an appraisal. There is often a difference between the public record data and an appraisal.
As long as they identify the source of the information, the listing can have any number. There is no requirement that public record square footage be used. I have seen square footage underreported on public records many times. The city could have forgotten to change the square footage after an addition or something.
The data on the MLS is not fake. An agent will take a look at the source of the information and decide if further investigation needs to be completed.