No, the symbiotic relationship means that the bacteria live in/on the plant, much like we have a gut biome of organisms that are discrete from us, but have symbiosis with us.

The article is about a bacterium becoming engulfed within an algal cell and slowly becoming an organelle of that cell.

This process begins as a type of symbiosis, but at the end of it you would no longer call it symbiotic, as it ends as a single organism.

The breathless yammering about "the last time this happened we got plants" seems a little much though. The last time this happened that we know of was 1.5 billion years ago and we got plants. The fact we've caught it in the act right now sounds like this might just be a much more common phenomenon than we thought.

The article says

In the 4-billion-odd-year history of life on Earth, primary endosymbiosis is thought to have only happened twice that we know of, and each time was a massive breakthrough for evolution.

It's possible the only reason we're aware of these events is because they were massive breakthroughs. After watching it happen, we may have more information about how to spot other times it has happened.

No plants are actually able to break down nitrogen gas. The bacteria that can break it down live in their root systems, not in the plant cells. The root systems are an ecosystem for the bacteria, much loke yoir moith and intestines are for useful bacteria. What this study describes is an algae cell incorporating a nitrogen fixing bacteria into it's cellular structure, which as far as we can tell has only happened twice in the history of life on earth: when a large bacteria incorporated a smaller one, thus creating eularyotic cells with mitochondria (which was our last common ancestor with plants) and when another eularyotic cell absorbed a photosynthetic bacteria, creating plant chloroplasts.