This statement is true. From the National Library of Medicine (https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5800290/)
I found the answer in an article published by the National Library of Medicine that was written by a group of psychology research specialists at BMC Psychiatry. This article is published on a United States government website. There is evidence to show a positive correlation between emotional support dogs and improvement in mental health among those with a mental illness. The article does state that more research needs to be conducted however there is existing evidence to prove this claim as true.
BMC PSYCHIATRY (NLM) - Quantitative evidence relating to the benefits of pet ownership was mixed with included studies demonstrating positive, negative and neutral impacts of pet ownership. Qualitative studies illuminated the intensiveness of connectivity people with companion animals reported, and the multi-faceted ways in which pets contributed to the work associated with managing a mental health condition, particularly in times of crisis.
This review suggests that pets provide benefits to those with mental health conditions. Further research is required to test the nature and extent of this relationship, incorporating outcomes that cover the range of roles and types of support pets confer in relation to mental health and the means by which these can be incorporated into the mainstay of support for people experiencing a mental health problem.