A soap opera is a program without a specific genre. There will be many storylines so of which will be dramatic (sometimes overly dramatic), some which are funny, sad or romantic. Usually, a soap opera broadcasts an episode every day and goes on for a long period of time, sometimes years.
A drama is usually a very specific storyline that deals specifically with emotional and dramatic themes. They will usually have a specific number of episodes or seasons.
In the clip, he is using the word soap opera to emphasize that her complaints have dragged on and are overdramatic. I think you could replace it with the word drama but it may lose some of the emphasis.