Thinking about the distinction between language and metalanguage may be helpful here. When one describes set theory as possessing a single binary relation denoted $\in$, one is operating at the level of metalanguage. Specifying axioms satisfied by $\in$ is at the level of the language. At this stage sets could be beer mugs as Hilbert famously said in a slightly different context.
Next, one assumes the existence of a model of the language, and interprets the meaning of the language, or more precisely of the theory expressed in the language, in that model (no more beer mugs).
In my experience, traditionally trained mathematicians (who have never taken a logic course) have great difficulty with the language/metalanguage and theory/model distinctions. This is because some of them tend to think of mathematics as "one great monolithic thing" and introducing such dichotomies goes counter to that philosophy. I don't think Paul Halmos ever overcame this.