I think this is not an attempt "to influence the crypto market", but some tricky person just created this token to sell it during hype. Elon Musk zero relationship to this Floki token.
This is a great example of how greedy people can be and how easily they can be tricked and controlled. This is so simple. Take what ever Elon says, create a token with same name and voila, you are locked and loaded for scam sale.
As you said, we don't know what they are really up to with this kind of strategy since they are prone to make things up when the crypto market has not flowed to how they want it. So, it's better to be safe than sorry because at the end of the day when the price of that new coin will reach its ATH, they will change their mind to manipulate its market then the one who will gonna lose their money are those people who invested late. However, if you are a risk-taker you can go with the flow and escape whenever you think a storm will come.