Why is there a cost of IOPS-month provisioned for activated ebs on stopped ec2?

Charges for Amazon EBS volumes accrue whether or not they are attached to an Amazon EC2 instance, or whether the instance to which it is attached is running or stopped. The provisioned IOPS is an attribute of the volume, not of the instance to which it may be attached.

To save cost on volumes that will not be attached for a long time, consider taking a snapshot of the volume and then deleting the volume. Snapshot costs are significantly lower than volume costs, and they are based only on size, not on length of storage. When you need the volume again, you can hydrate it from a snapshot - even in a different Availability Zone than the one in which the volume was initially created.

As to answer your question "why there's a monthly cost for IOPS?" - that's just the way the pricing for some features and services works, where you pay for the provisioned resources, rather than used resources. For example, you pay the same way for the EBS storage - you provision a certain amount of GB/TB for your volume, you may be actually using only a portion of that provisioned storage (there will be less data on the volume), but in the end you will pay for everything that you actually provisioned. Same with provisioned IOPS - you pay for what you provision regardless of whether the volume is attached to an instance or not, or whether the instance is running or not.

There are some services where you actually pay for what you use instead of what you provision. One example - DynamoDB services. You can create DDB tables in provisioned mode - in this case you'll be paying for provisioned Write and Read Capacity Units; or, you can create a table in on-demand mode, in that case you pay for invididual requests ("pat per request").