A huge part of the above-water corporate iceberg is the people and your interactions with them, so the company does take on a proxy "emotional signature" based on with whom you interact and the context of the situation. I don't see how a computer program trained on the human knowledge corpus does anything more than parrot observed behaviours without the backing biological systems. Mirroring pretty much the opposite of genuine emotion.
What's so special about those "backing biological systems"?
Claude might have an emotional signature that is all cuddly touchy-feely in an abstract, intangible, disembodied way. But.. Anthropic - as a corporation - will still have that deep, dark, insatiable desire to rape your wallet.