Well, sure... There's a lot going on in your question. Let's break it down a little.
The Priority Field is a global priority field within JIRA. This can be configured under –
You can add new values or set a default value. Please read more about Priorities here – Defining Priority Fields.
If any specific field is not editable, or if the field is not present during the Edit operation, then the person attempting to make the change does not have permissions to Edit or the field is not present on the screen assigned to the Edit Operation within your project. Alternatively, you may have hidden the field on the operation, under your own view settings. Please read more about Editing issues.
Now, there's a little oddity with JIRA. System fields (opposed to Custom Fields) are visible when viewing an issue, even if they aren't present on the screen assigned to the View Operation. This means in order to hide a default field, you should remove it from the View Operation screen and set the visibility in the field configuration to Hidden. You can read about hiding fields under Specifying Field Behavior.
Yes... That is the nature of the Priority field. It is global.
If this is a problem for you, your only solution is to create your own as a custom field and hide the default priority field via Field Configuration. You will not be able to replace it's functionality (icons that display in certain areas).
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