The issue is not related to PackageReference vs packages.conf. With old-style project files, both ways to reference NuGet packages work.

By default only .Net Standard and .Net Core projects use this file format. Since System.Data.SQLite.Core 1.0.110 the build folder in the NuGet package no longer contains a targets file for netstandard2.0 (it did exist in 1.0.109).

It looks like this is more of an edge case that might not be officially supported. I have therefore reduced the severity to "Minor".

Workaround for me is to use the old project file format.