Interesting question! While it doesn't exactly address the topic of your question, I think the paper Is painting by elephants in zoos as enriching as we are led to believe? might have some insight.
I originally found this paper from the animal-made art wikipedia page, which states that:
The elephants draw the same painting each time and have learned to draw it line-for-line.
Although I couldn't find that claim in the paper they cited, it does seem like this is a highly structured activity (as opposed to a more free-form "art"). For example, from the above paper:
Painting involved following instructions from the keeper and performing the task in front of an audience. ... The brush was then handed to the elephant, which it then held with its trunk. Instruction was given by the keeper to the elephant when to start and stop painting, and when to give the brush to the keeper for colour changes. ... Each session lasted between 2 and 5 min, and was a highly structured event. For example, when the elephant was not holding the paintbrush and awaiting the next instruction it was directed to rest its trunk tip on the ground.
There might be more information in this book, Cerebrum 2009: Emerging ideas in brain science, as it's cited a lot in the introduction, but it doesn't seem to be available online.
Overall, it doesn't seem like the elephants have much agency in what/how they paint. As the authors put it:
[T]he results indicated that elephants gain little enrichment from the activity of painting. Hence, the benefits of this activity appear to be limited to the aesthetic appeal of these paintings to the people viewing them.