There is a good chance that your .NET Core 3.1 library will run without any compatibility problems as part of your .NET 6 application.
Quoting the .NET fundamentals article Changes that affect compatibility (emphasis mine):
Throughout its history, .NET has attempted to maintain a high level of
compatibility from version to version and across implementations of
.NET. …
Along with compatibility across .NET implementations, developers
expect a high level of compatibility across versions of a given
implementation of .NET. In particular, code written for an earlier
version of .NET Core should run seamlessly on .NET 5 or a later
version. In fact, many developers expect that the new APIs found in
newly released versions of .NET should also be compatible with the
pre-release versions in which those APIs were introduced.
However, compatibility is not guaranteed. The article goes on to say:
This article outlines changes that affect compatibility and the
way in which the .NET team evaluates each type of change.
Understanding how the .NET team approaches possible breaking
changes is particularly helpful for developers who open pull
requests that modify the behavior of existing .NET APIs.
Furthermore, in an answer to the question Clarification on backwards compatibility of .NET Core, a member of the .NET runtime team says:
We do not guarantee 100% compatibility between major versions. This is true for both ASP.NET Core and the runtime itself. We
intentionally make breaking changes where we believe that they are
necessary to move the platform forward and the cost of the .NET
ecosystem adjusting to them is low enough.
Breaking changes that could potentially affect the compatibility of your .NET Core 3.1 library are those that are documented for .NET 5 and .NET 6.
But, if you are not experiencing any problems with your .NET Core 3.1 library, it would appear that none of the documented compatibility problems apply.
Ultimately, of course, you'll have a higher degree of confidence in the library when your vendor provides one that has been updated for .NET 6.
Finally, with the introduction of .NET 5, there became a lesser need for .NET Standard:
.NET Standard is a formal specification of .NET APIs that are
available on multiple .NET implementations. The motivation behind .NET
Standard was to establish greater uniformity in the .NET ecosystem.
.NET 5 and later versions adopt a different approach to establishing uniformity that eliminates the need for .NET Standard in most
scenarios. However, if you want to share code between .NET Framework
and any other .NET implementation, such as .NET Core, your library
should target .NET Standard 2.0. No new versions of .NET Standard will
be released, but .NET 5, .NET 6, and all future versions will continue
to support .NET Standard 2.1 and earlier.