Whether it is true or not makes little difference when you are not supposed to share any PI or PII with any AI anyway.
Cautiously use AI for help you actually need but never for mental health or anything personal like that. Use it as a tool for utilitarian purposes only.
And yes, any AI will indeed store any info fed into it. There are or may be some exceptions via API if you use another front und for the service to select the option of not using your queries for training data but only to provide answers too. But even then, I’d not trust it fully.