You really have made a rod for your own back by using two separate, Windows and Linux, Libraries.
First, the <Date>3178</Date> you listed isn't correct. That is 12th September 1908 (09/12/1908 in US format). You missed a zero on the end. 31780 is 3rd January 1987 (01/03/1987 in US format). As this is a scanned image, you probably did enter 1987 as the [Year] in Windows MC, and at the same time MC updated the [Date] field to 1987. MC will do that.
The Windows and Linux file system dates of 1 Oct 2005 would be the file system "Date Created" or "Date Modified" date. The [Date] tag is completely different. Have a look at all the date fields supported on the file system and in MC.
Next, you are using two different Libraries, so you can have two different values in the [Date] field, one in each Library, unless you are forcing updates to the separate Libraries. So you changed the [Date] tag to 1987 in Windows MC, but either;
1. You don't have Windows MC set up to write the [Date] field to the file tag (that would be unusual because I'm pretty sure by defaults MC is set up to write that field to the tag in the file), or
2. Windows wrote the updated date to the file, but you don't have Linux MC set to "Update for external changes" in Auto Import and have Auto Import running.
On Linux, select the example file, right-click and select "Library Tools > Update Library (from tags)". The Linux MC [Year] will probably change to 1987, because the <Date>31780</Date> tag in the file would indicate that the correct, 01/03/1987 value, is in the file.
If that worked, you will need to do the same for all problems files, and then set up Auto Import to run and update for external file changes.
In case you are wondering, I call metadata in a MC Library "fields" and metadata in a file "Tags".
PS: Search for "Tag Dump" in the forum, and then set that up in both your MC Libraries, and use that to check what dates are actually in the files, if you haven't already done that.