How does the Tech Support know my exact location given my location services is turned off?
There could be multiple sources that Apple has received your location from.
Even if you do not think about it, your IP addresses is actually transmitted to the web server (Apple's server) every time you load up a web site. In many cases an IP address can pinpoint your location to a specific city or similar. This would be enough for them to able to suggest a store in a specific zip code.
In addition WiFi-enabled devices without GPS are still able to find their location from other sources - for example the names (SSIDs) of WiFi access point the device &quot;sees&quot;. When it is moved around, it is possible to assemble a list of known access points which can then be translated into a position by an online server. For example the cars that Google uses to gather data for Google Maps also collects WiFI SSID information for this purpose.
If you're especially worried that Apple support knows your location, I would just ask them where they have your address from. If you're located within the European Union, you have rights based on the GDPR that would ensure that they're obligated to tell you that they have this information on file.