The above solution is good if you need it as a temporary solution…
For a long term, change your user settings in:

File -> Preferences -> Settings

Scrool Down until you will see:

Files(14) - number might be different

There you would see text like:

// Configure file associations to languages (e.g. “*.extension”: “html”).
These have precedence over the default associations of the languages installed.
“files.associations”: {},

Press edit → it should copy settings automatically to the right screen where you have USER settings.
Change to:

“files.associations”: {“*.html.eex”: “html”}

Save and Close File with settings

Test that it works

If you use the vscode-elixir-ls extension, note that EEx-related settings may need to change due to recent updates. With the latest version of that package, capitalize the second E:

"emmet.includeLanguages": {
  "HTML (EEx)": "html"
}

I was able to get emmet working in VSCode with the following setting:

"emmet.includeLanguages": {
    "HTML (Eex)": "html"
}

I wish support for vscode was better. I have no intellisense for elixir, there’s no formatter to auto-indent the html and no autocomplete for phoenix functions that are available in the template.

Language tags have been changed with the release of 0.4.0 to follow visual studio code guidelines, which has been mentioned in the changelog:

VSCode potentially breaking changes:

• Change language id to be lowercase kebab-case in accordance with VSCode guidelines. This also fixes an issue displaying the elixir logo for html.eex files. (thanks Matt Furden) #87
  • This changes the language id’s EEx → eex and HTML (EEx) → html-eex

[…]