Unlike Tableau Professional, Tableau Public does not offer the option to use custom shapes as marks, making it difficult to create more visually interesting tooltips.
Follow along with the video below to see how you can create custom shape tooltips in Tableau Public, or read through the steps in this article. (Each written step has time stamps corresponding with where you can find them in the video).
In a blank sheet, drag in a measure and change the mark type to a square. You can also change it to a circle if the custom shape you would like to use is circular (0:06).
Then switch the measure to color, and change the fit to "Entire View" (0:17).
Lower the opacity of the mark to 0%, and make sure the shading on your sheet is set to "None" (0:23).
On your dashboard, drag the sheet with the mark on top of the image you would like to use as your custom tooltip shape. Remove the title and legend (0:34).
You can click inside the border of the sheet to toggle the transparency to make it easier to size correctly (0:48).
To edit the text that pops up, go back to the sheet with the mark and click "Tooltip" under the "Marks" menu (1:01).
Enjoy your custom tooltip!