Any template tag rendered by EEx and served by Phoenix is going to be interpreted by the browser as a native content template element which is an entirely different beast.
Even if Vue can use the template content client side - you should aim to avoid any approach that requires that you use the Full build:
One reason why I brought up those issues is that Vue.js is typically used to build an SPA.
In that case it is common to serve the SPA completely decoupled from Phoenix - which also leaves the option open to use Nuxt.js for SSR to pre-render Vue.js to improve client-side performance. Phoenix then acts simply as a JSON or GraqhQL server (with Absinthe) for the SPA.
Going down the Vue.js SPA road there really isn’t any use for EEx templating. EEx is more useful for building dynamic web pages. Now it is conceivable to use dynamic pages that include a component that uses Vue.js but that use case doesn’t seem to be all that common.