How can I make my RFID reader restrict itself to the tag in the closest proximity and read only this tag?
An RFID reader can't guess distance, and especially not until trying to "read" a tag.

Some RFID standards (there's different ones, do your research what fits your purpose best) allow for collision avoidance; but those basically "partition" the ID space to "detangle" interaction. 

But physically, an RFID reader has no way of knowing which tag is closest. Sure, it might have an option to guess based from signal strength, but that's pure guesswork and won't work well with different tags in different orientations and different media in between reader and tag.

If "closest only" is a strict requirement, you'd probably have a use case for optical technologies like bar codes or text labels.