[This was long, so ... PART 1]
I'm going to start off by saying that this information is not documented anywhere — it is only what I've been able to figure out and so my explanation is solely based on observation and trying to "make sense" out of it.
Link is an incredibly powerful behavior but it can seem rather "quirky" from time to time as in this instance.
I started realizing quite awhile back that Link behaves according to "like parameters". By that I mean, if you have two rectangles and you link the same (like) parameter from one to the other, then you get the full complement of features that Link offers. If you cross over to an object that has different (behind the scenes) setups, then Link may or may not offer all of it's parameters for use.
When dealing with Text objects, much of what goes on behind the scenes is very much different than simple width/height measurements. There's things like Tracking, Kerning, Font, Styles, etc... and a whole collection of "metrics" (ascent, descent, etc.). A font's metrics start off in an imaginary (internally defined) grid that is applied to its destination. (I know this because I also design fonts). In the case of 3D Text, this internally defined grid helps establish its resolution (the more "intersections" in the grid, the more refined the 3D Text will be.)
So, back to your issue.
It looks like you have a Rectangle and some text and I'm guessing you want the Rectangle to be a minimum width and as Text is added to the Text object, you want that Rectangle to grow with the width of the text.
As you've discovered, when you Link them up for width, there's no minimum source value to use. There is a mismatch in parameters *even though* both deal with width.
Whenever you run into a condition like this, and this won't be the only time (believe me!) — the Numbers Generator is your friend.
Numbers is a generic number. It will match any kind of parameter (more or less — there are caveats). You will need to link objects values through a Numbers Value. What you have to watch out for is "resolution". Some parameters are represented by factors of 10 [and sometime π] different than they appear in the Motion interface (this will be one example).
A quick note about Numbers: it's default condition is to Animate (checkbox). You will never have access to the animated "value" of a Numbers generator. In order to use the Value parameter via Link, you must always uncheck Animate.
To set up something like the following:
Draw your Rectangle and make the width fit nicely with the text for say 4 characters. The text in the example below is emulating a "Type On" with a Sequence Text Behavior using Scale.X set to 0 and Spread set to 0. (End Offset used to control speed of "write on").
Set the Rectangle width to about 240 (or wide enough to fit 4 or so characters — I'll just stick with 240 for this example).
Add a Numbers Generator (you can turn off the visibility before going "live".) Turn off Animate and add a Link to the Value parameter.
To the Link Source Object, drag and drop the Text object and set the Source Parameter to Object Attributes > Size > Width.
To the Rectangle Shape Geometry Size Width parameter add another Link. For its Source Object, drag in the Numbers Generator and set the Compatible Parameters to Object.Numbers.Value.
Notice the Rectangle goes very wide. In this case, Numbers Value represents a "resolution" 10 times the Size value, so set the Link's Scale to 0.1 (to bring it back to "reality"). Set the Apply Link When Source value above minimum [which will appear in this circumstance] and set the Value Min to the original width of the Rectangle (about 240).
As you type in characters to the text object (or animate them as I did) the Rectangle will begin to widen once the minimum size of text reaches 240 pixels.
