> Do we see signs of reasoning or is it anthropomorphism?

This is the part that so many folks just don't seem to understand (probably because it's been labeled as "thinking" or "reasoning" mode, and people assume that words have meaning). It's not reasoning or thought. It's spewing tokens pretending to "think", but it's actually just generating extra "context" to help the final answer be more coherent. The model isn't doing anything it doesn't already do. It's just doing more of it to improve the quality of the final answer displayed to the user.

Yes, we do see signs of actual reasoning, see the papers linked in the article. (There are many others too.)

Yes, we have a tendency to anthropomorphize, but (most) researchers are aware of this.