Axler serves as an adequate first introduction to linear algebra (though it is intended to be a second, more formal, pass through. Think analysis vs calculus), but it isn't intended to be a first introduction to all of formal mathematics! A necessary prereq is understanding some formal language used in mathematics- what unique means is included in that.
Falling entirely back on physical intuition is fine for students who will use linear algebra only in physical contexts, but linear algebra is often a stepping stone towards more general abstract algebra. That's what Axler aims to help with, and with arbitrary (for instance) rings there isn't a nice spacial metaphor to help you. There you need to have developed the skill of looking at a definition and parsing out what an object is from that.
Falling entirely back on physical intuition is fine for students who will use linear algebra only in physical contexts, but linear algebra is often a stepping stone towards more general abstract algebra. That's what Axler aims to help with, and with arbitrary (for instance) rings there isn't a nice spacial metaphor to help you. There you need to have developed the skill of looking at a definition and parsing out what an object is from that.