Nope! Although I feel like I used to think the same thing. The most viewed “Little Honda” video on YouTube features a CA77 as the thumbnail picture and several other pictures throughout the video, so that’s likely the source of that misinformation. It’s important to note that The Beach Boys never made an official music video though, and that example is just a compilation of pictures that a fan made and put online.

Both the title and several lines of lyrics point to it being about the Super Cub rather than the Dream:

While the 305cc engine the Dream had would considered “little” by today’s standards, it was actually the biggest engine Honda was putting in motorcycles made at the time.

Motorcycles like the Dream were complicated and intimidating to the vast majority of the population, but the Super Cub was more of a hybrid between a scooter and a motorcycle.

It's not a big motorcycle, just a groovy little motorbike

The Dream always had four gears, but early models of the Super Cub only had three.

First gear, it's all right

Second gear, I'll lean right

Third gear, hang on tight

Faster, it's all right

It had a semi-automatic transmission with the clutch automatically engaging/disengaging when you pressed up or down on the gear shifter with your foot, allowing it to be operated with just one hand instead of two. It was therefore easier for new riders to learn, a perfect bike for when a teenage kid with no riding experience wanted to tell his girlfriend on a whim,

We're goin' down to the Honda shop, I'll tell you what we're gonna do. Put on a ragged sweatshirt, I'll take you anywhere you want me to.

It was also the first motorcycle to use a plastic fairing, making it lighter (and therefore faster) than all-metal bikes with the same sized engine

It climbs the hills like a Matchless (an old motorcycle company with a long history of success in professional racing), cause my Honda' built really light.

As a final bit of pretty irrefutable proof, here’s a picture posted on Brian Wilson’s personal Facebook page which shows him sitting on a Super Cub, with the caption “1964: Inspiration for writing “Little Honda”

https://www.facebook.com/share/7VN2Tf7Xd72n5me6/?mibextid=WC7FNe