No Good Neighbour Hotels and Motels do not get early entry or any special perks. The Good Neighbour Hotel means that the hotel has paid a fee to Disney to get included on the Disney website booking system and be an authorized park ticket seller.
Anything in Fantasyland, Tomorrowland or Adventureland / New Orleans Square from park opening to 11am.
Staying at a good neighbor hotel near the harbor entrance typically significantly less expensive than the 3 Disney hotels. Most of the hotels directly across the street on Harbor Blvd are actually a shorter walk to the Esplanade than the walk from PPH or DLH. Staying at a nearby good neighbor hotel and rope dropping the park that does not have early entry and using the money saved to purchase Genie + and ILL for RotR and RSR is actually a less expensive option than staying onsite. You will also have a less stressful rope drop if you don’t attempt RotR or RSR and focus on the more reliable rides at rope drop (I assume you are interested in rope dropping if you are inquiring about EE).
As others have noted, no, that perk isn't currently available. But you aren't imagining things: there used to be a program called Magic Mornings whereby people who weren't staying at a Disney hotel but had three-day or longer Disneyland tickets could go in early one morning. Alas, that program was not brought back when Disneyland reopened after its COVID closure. I wouldn't make a large bet that it ever will.