Claude has this by way of projects, you can set instructions that act as a default starting prompt for any chats in that project. I use it to describe my project tech stack and preferences so I don't need to keep re-hashing it. Overall it has been a really useful feature to maintaining a high signal/noise ratio.
In Github Copilot's web chat it is personal instructions or spaces (Like perplexity), In CoPilot (M365) this is a notebook but nothing in the copilot app. In ChatGPT it is a project, in Mistral you have projects but pre-prompting is achieved by using agents (like custom GPT's).
These memory features seem like they are organic-background project generation for the span of your account. Neat but more of an evolution of summarization and templating.