There's no definitive way to tell if some code is written by AI or not; thus their statement doesn't have to be true. Usage of AI itself is nebulous, it could mean anything from OpenClaw-style automated agents to someone prompting an LLM via chat to an engineer manually writing code after wasting hours trying to get an agent to do it (that still counts as "usage", even if not ultimately productive).
The market currently rewards claims of usage of AI, so everyone is claiming to be using AI. There is no way to prove one way or another, and the narrative will be changed down the line once the market swings.
When it comes to productivity claims, I have yet to see it. If AI is truly providing significant, sustained productivity improvements across the software development lifecycle I'd expect products to be better, cheaper, or get developed faster (all of which happened with other industrial breakthroughs). I do not see that in software at large and especially not in Spotify's case.