Thanks for tagging me @JoyHawkins
The favicon showing will be displayed next to chain/brand results. This is Google Maps' way of grouping the chain locations together to give you results just for the brand. This would be similar to a branded map pack on search that limits the view of reviews. When you click on the group you will see only the results for that brand.
Unfortunately, if Google does not have you identified as a chain/brand in their knowledge graph, you likely will not be able to achieve the same experience.
Thank you so much for this! One additional question - how would one go about helping Google identify that a business is a chain/brand in their knowledge graph? Specifically speaking about one business in particular, Lucky Lion in Portland. Any help would be greatly appreciated!!
There isn't a straight forward or precise way. A lot of it has to do with consistent display and indexation of your listings across the web. As well as clean and structure markup on your site that matches your local listings. And yes as @klharris08 pointed out, you can submit Chain Name via API. This should be consistent on your digital properties.