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Your default system settings will launch your DVD Player app when a DVD movie disk is inserted.
That will make the DVD Player icon appear in the dock, if it is not already there.
When you quit DVD Player (not simply by ejecting the DVD), then the DVD Player icon will disappear from the dock.
More relevant to your question -
No, there's nothing that will appear in the dock that tells you that a DVD is is the drive, unless you have your system set to do that.
A DVD disk (regardless of whether it's a movie, or just a data DVD) WILL normally appear in the sidebar of a finder window - again assuming that you have the system set to show CDs & DVDs in your finder sidebar (from your Finder Preferences.)