"As a language model" disclaimers were certainly explicitly trained into chat models in the early days. It's quite possible that it has since bootstrapped into a "fact" that later generations of LLM know about how LLMs speak, in which case they may be doing it even without any posttraining that encourages it.

These formulations have been selected by reinforcement learning. People who aligned the LLMs chose this over alternatives.

You know when chatbots ask you which answer you prefer between two. People tend to chose the "as a langage model..." one, so it stuck.