https://docs.aws.amazon.com/whitepapers/latest/how-aws-pricing-works/estimating-amazon-ec2-costs.html
About Elastic IPs, we read (emphasis mine):
"Clock hours of server time: Resources incur charges when they are running—for example, from the time Amazon EC2 instances are launched until they are terminated, or from the time Elastic IPs are allocated until the time they are de-allocated. "
(I read this to mean EIPs are not free - charges are incurred for allocated EIPs whether the EC2 exists or not.)
And later in the page we read:
"Elastic IP addresses: You can have one Elastic IP (EIP) address associated with a running instance at no charge."
So the way it's written, it can be confusing. At face value, these two quotes appear to contradict each other. IIRC, the first associated IP is free, but if it's not associated or not first, then there are charges - which isn't so clearly stated at the web page, imho.
HTH?