I am going to set this as an answer since, after Hans Passant's very helpful comments and subsequent research, I think it's pretty definitive that this can mostly only be worked around, not resolved. (As possible exception will be discussed at the end of this answer.)
As stated in those comments, this error originates from a linker module called al.exe that is utilized by MSBuild for certain project configurations (more on that later). This linker can be found in a couple places, but for me it was being called from C:\Program Files (x86)\Microsoft SDKs\Windows\v10.0A\bin\NETFX 4.8 Tools.
MSBuild can handle long file paths as long as Windows is configured to do so (by enabling long paths in the Group Policy Editor or by modifying the registry directly). However this al.exe module cannot. And as far as I can tell, there's no way to force it to do so. So if your build tool chain requires this al.exe module to be used, you're kind of SOL.
For my particular situation, for a job in Jenkins that was failing because my paths were too long, I worked around this by changing the workspace of my job. So now instead of the default of something like D:\Jenkins\workspaces\[product]\Releases\[product_version], I changed it to D:\j\dd0_1c, which is an encoding that makes sense to me. This shortening of the folder path avoids any subsequent file paths from exceeding the limit of 260 characters. It's not a satisfying solution, but it works for my particular situation.
I did mention that there was a possible exception to all of this: if you can get MSBuild to avoid using al.exe altogether, then you can avoid this error.
I don't know all of the scenarios or workflows in which MSBuild utilizes this module, but I do know that it does get utilized when your application has localized resources, and somehow, some way, MSBuild uses al.exe when it is generating those resources. This was exactly my scenario, and I found this page and this page describing how you can reconfigure your localization projects such that MSBuild does not utilize al.exe. I did try the steps described in these pages and was able to verify that I no longer got this ALINK error from al.exe. However I never got my project to fully build since this reconfiguration caused other build errors to crop up. So in the end, for the sake of expediency (and because it was cleaner than performing a major refactor of my code), I went with the Jenkins workspace workaround.
However, it is interesting to note that you can get MSBuild to avoid using al.exe as long as your project is conducive to the solution given in those two links. So hopefully, if someone runs into this same issue, they might have more success than I in utilizing this method.
