Is there is a possible way to make a "text style" act on each line of "Graphic" differently ?

Got it. And you are explaining very well.

This is an increasingly common method: styling captions by the word or animating (pop-on) the caption. And more common for what I assume you are using: a vertical frame.

But I see it more with single lines, and that is easier. I think I might make them one line, or change something else that doesn't require having each line in its own layer. But let's see... If you have many captions, this will be too labor-intensive. If short, it might work. 

Keep the current captions track (for timing as 2-line captions), and create captions from the transcript tab with different parameters so they are one-line captions (the first is your original caption 1 top line; the second is your original caption 1 bottom line). Check them to determine whether they are okay, or if you need to edit any of them so the top line/bottom will match your original.

Or you can duplicate your original caption track, split each caption, and adjust timing and delete text so you now have 2, single-line captions for each original. 

Duplicate that new caption track twice (create a new track, select all captions on the track, and Alt+Drag to the new track). You now have 4 caption tracks. The original 2 line track. A single-line version as a backup (when you upgrade the captions, they are removed from the caption track). And 2 duplicate single-line tracks.

Select the captions on one of the duplicate tracks. Upgrade captions to graphics. Repeat so the next track goes onto a different video track. So you now have, for example, one complete set of single line captions on V3 and one on V4.

Now the bad part. We need V3 to be the top line of captions and V4 the bottom. So delete V3 caption 2, 4, etc. Delete V4 1, 3, etc. Now Move the captions on V4 to line up with the beginning of each caption on V3. And then drag out the first caption on V3 and V4 to match the timing of caption 1 in the original 2 line captions (that we left on the original caption track). We now have 2 line captions in separate graphics instances.

Adjust font color, position etc of the first V3/top line caption. Save a style, and note where it is in the Project Panel. Select all captions on V3. Drag that style from the Project Panel to them on the timeline. Repeat for the first V4/bottom line captions.

Does that work? I hope someone can suggest a better way....

Stan