It's true that a JavaScript hash can be accessed using either dot . or array [] notation. If you have a literal name, you can use dot notation this.$.some_id. If you have an indirection, like this.id = 'some_id', then you can use array notation this.$[this.id] to find the same value.
The tricky part is that Polymer only populates $ array after first stamping the template, which happens before ready. If you had an external binding to this.id, this.$.[this.id] would work, but since you are setting this.id in ready it's too late for the $ convenience.
In this case, you can instead query your shadowRoot directly:
this.shadowRoot.querySelector('#' + this.id)
Pro tip: at some point a subclass may supply a new template, in which case this.shadowRoot will point to the new shadow-root and not the superclass version. For this reason, it's best to install a named div you can query against, e.g. this.$.id_div.querySelector('#' + this.id').