Your understanding is correct that, unless indicated otherwise, all files in a repository are covered by the same license (3-clause BSD in this case).
The BSD license allows you to copy and modify the project (which includes adding it to your own project and then throwing everything out except the icon you want) on the condition that you keep the copyright statement and BSD license intact for the portion you copied.
This means that you must make it clear
- that you copied the icon
- who owns the copyrights to the icon
- that the icon is licensed under the 3-clause BSD license
- that this differs from the copyrigh and license on the rest of your project