As long as the same paragraph styles are the same name, but they can be different settings - then using Library would be a way to go.
Drag your text frame in
Document 1..................Document 2
Phone Number..................Phone Number
As long as the style names in Document 1 and Document 2 are identical - then dragging them from the library should work.
Well actually - that would mean you'd have to update your library each time and replace all the information across all documents when the details change.
Snippets would present the same problem.
Text variables would also be the same problem and harder to update.
Seems the best option is a linked MS Word file.
One change to the MS word file and you'd receive notification of a broken link in InDesign - Prompting you to apply the correct styles to the text.
I'm not sure - but I think if you use the same Style Names in MS Word and then link it to InDesign - once the style names are identical I think it will use InDesigns styles.
Unfortunately, using the Library does not retain a link between the text everywhere it appears. I am looking for a solution that works similar to placing an InCopy file as a link. If I update the text in one place (Document 1, Document 2, or the InCopy file), the text updates in all locations. The problem is that I don't want to be tied in to using whatever paragraph style is assigned to the text. For example, in one document the text might use a "Header Sans Blue" paragraph style, and in another document it might use a "Body Serif" style. It would not make sense to name these styles the same because they describe fundamentally different paragraph styles.
Is there maybe an XML solution, date merge, or some other tool that could be used for syncing only text and ignoring styles?
Use an InCopy story.
Bob's solution is the best workaround for this kind of situation. Sounds like you want to manage content and minimize manual reuse of text, but don't have any content management tools. This is exactly the kind of thing that people do with, say, XML and FrameMaker and their preferred flavor of database. And, if you are using InDesign to manage your content in this way, you are essentially driving nails with a crescent wrench. You can do it, but it's not a particularly elegant solution.
I advise people with questions like this to check out DocsFlow or WordFlow from Em Software. I'm not a user, but I've played around with it and it covers a lot of ground for high-level users of InDesign and word-processing tools who may not have other tools to manage textual content without a lot of manual pushing-about of text.
That being said, in all the cases where I don't use translation memory tools to do the heavy lifting for me, I do as Eugene has described; I save content in Word, place with a link to the Word file, carefully map the Word styles to InDesign styles, and save the style mapping(s) so that updates to the Word file lead to almost-automatic updating of the target text in InDesign. Because Matt's case is one in which the styles would have to be ignored, you couldn't just use the same style names in Word as in ID - he almost certainly has cases where the same text is styled in multiple locations with different style names.