I think Elon and Doge have a lot in common so what Elon does has an impact on Dogecoin. Every time Elon tweets the price of Doge increases but the current market supply is high but the price is very low. The market is not fixed no matter the tweet tries to increase the price.
There could be no connection between Elon Musk and Doge coin so far maybe he is just an investor same way everyone is. The only reason is just due to his influence. Elon has a bigger influence on making even project with less value succeed with just a single post. If he would make a post on any other Altcoins, same way the price of the coin will pump.
If he doesn't have any connection then why does he call himself doge Father? He also talked about restarting the doge coin development to make doge from inflationary to deflationary. He must have seen a way to make money out of it and he is taking his chance. I know why he picked doge coin because it was easy to accumulate so much coin from the market unnoticed due to its huge supply and he knew his influence would pump dogecoin to a new ATH. He is really a smart man and he knows how to make money.