IMO the correct specification is more important than Brand. There is a lot of 'own label' packaging in the lubricating oil industry, and there is only a limited number of oil fields ,each producing a different quality of crude oil that has to be refined to a consistent specification of base stock, by a limited number of oil refineries. A few major oil companies may produce their own unique product with their own additives with fancy names, but many oil blenders use additives made by massive internationally known chemical companies such as BASF.
For Instance 'Comma oil' is seen as a bit of a budget brand in the UK. Originally independent blenders, they were then owned by Esso, then Mobil Exxon, and currently by a Brazilian Company, who produce Mobil products in the South American region. Trucks leave The Comma site in uk with both Comma and Mobil logos. Comma claim to have packaged 'own label' oil for Toyota, Landrover etc. So is this Mobil oil, Comma oil, or specially made oil?
Or is it the same stuff with customers paying a premium price for a well advertised name?
If Honda factory fill a Jazz with synthetic 0W20 oil I cannot see why anyone would want to revert to anything less to save a few quid. 5W30 may be an acceptable substitute, on some older models or high mileage cars but its a thicker oil that may not circulate as easily, especially in colder weather.This will increase fuel consumption.
Honda dealerships may use various brand names, depending on who gives them the best price. But it should be the correct quality and specification. Some may be Suzuki etc dealerships as well as Honda and may share oil from the same drum.
I can see the appeal of providing your own oil to a small independent. If they have a drum of 5w30 in the workshop that dispenses a measured dose they may be tempted to use that, rather than go to the trouble of sourcing the correct stuff.