Data normalization (be it rarefying or another technique) is a separate process from estimating the effective number of species in a sample (alpha diversity values).
There is not one normalization technique, and different techniques are required depending on what you're trying to do:
If you're following along a classic tutorial, it's likely that you first rarefied your data, then calculated alpha diversity values. That's one way to go about normalization, though there are many others. See the recent SRS tool, for example. I would suggest that your linear model uses the same normalized data (in SPSS) if you're comparing to your KW result.
Minor clarification: my earlier response assumed QIIME 2 alpha group significance was running one of a range of possible post hoc tests, but it appears it's running KW for each group separately? I defer to the developers and users of the forum for the appropriateness of such an action, but would encourage you to apply a Dunn's test to see if your pairwise comparisons match what is provided with the QIIME approach. In particular, any pairwise test needs to be mindful of multiple test corrections, which a post hoc test like Dunn's applies (usually automatically). With just 3 groups, you have just 3 comparisons, so it might not matter. But if you have, say, 7 groups that's k(k-1)/2 21 tests...
The labels aren't incorrect insofar as they reflect your observed pairwise pvalues: the question is whether those pvalues are derived by the appropriate pairwise test, and whether they should be adjusted for multiple testing.