And most importantly the gradient has a length (size) attribute which unfortunately is not in the Gradient panel and is also not accessible as a numeric value but only by using interactively the gradient tool (G) and pulling the end point representing the length of the gradient. If you don't want to copy it with the eyedropper or as a style, I don't know of a way to accurately measure the length(size) of the gradient but eyeballing the ruler.

I have no idea what are you trying to achieve but it doesn't make sense to have the rounded rectangle with a gradient and a circle in front of it with a different gradient but trying to match both in the way you described. If you want to match the gradient of both objects, then select both objects, make sure the gradient is set to radial, take the Gradient tool and first drag away from the objects then drag again from the center of both objects to the end of the rounded rectangle and you will have a match - the circle will not be distinguished. Is this what you want to do?

Can you please, show a screenshot comparing how the gradient tool appears when all objects,  yours and referenced objects are selected?

What is the purpose of showing us inflated circle instead of just simple circle, box, etc? If you have a problem in general you should start with the most simple example.

Also if both  objects have the same gradient the Gradient panel will not show a question mark.

May be the best way of figuring out your issue is if you can share the file. I have to go now and won't be able to check it soon but I'm sure others can help.

I notice that in your screen that you are actually applying the gradient to a circle and then converting the circle shape to a rounded rectangle is that not correct?

Can we see your appearance panel and the other settings for the two rounded rectangles.

Because this is what would happen if it were an actual rounded rectangle.

Also I do not know how you are capturing the actual annotator tool unless you ae using a video still I am curious about this.

But as you can see there clearly other issues involved here and not just the gradient fill that could play a role in how the gradient is being msnifested and displayed.

Please give more information.