Currently, there are only limited HTML and CSS elements you can change regarding the design or looks of the Dialogflow Messenger chat window.
As for the floating feature, currently, it seems that this is not supported in Google Sites as also discussed in this StackOverflow post.
However, you can create a section for Dialogflow Messenger only and adjust the position of Dialogflow Messenger window/icon manually in Google sites. To do so, you may follow these steps:
At the right side of Google Sites page, click Insert and then Embed.
Click Embed code.
Add the Dialogflow Messenger embed code, then click Next.
Click Insert.
Move and adjust the position manually of the DF Messenger icon in the page
To publish your changes, at the top right, click Publish.