Jira is not a file sharing system, it's an issue tracker. The project header doesn't have a lot of configurable items, it's mainly for configuring the project. The best you could do off-the-shelf is either say "issue-1 in this project has all the attachments I want to share" or put a url pointer to a file sharing system in the project description text (Most places I've worked have a matching Confluence project and put the information/attachments there)
There's a plugin to provide project level fields, but I'm not sure it includes a way of handling files at a project level.
The plugin mentioned lets you add project level fields to your JIRA projects. Check out https://marketplace.atlassian.com/plugins/com.deiser.jira.profields
I am not sure it allows you to add "attachments" to the project as you looking for but you could possible take up this discussion with the vendor of the plugin.
Meanwhile i vote for Nic's suggestion to add a ticket containing the uploaded files for the project and link to it in the project description for example.
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