Report - Projects a User has Manage Access to

I have what could be a start of it...create a project report with project users ID equal [person's name in question] and can also add a filter for project owner ID NOT equal the same person.

From there having a hard time pulling in a column to show what level of access they have on the projects but this is a start to show you what they have access to and are NOT the project owner.

this would probably be a project report with an exists filter that is filtering on access rules.

You would start off with a regular report filtering for "project owner ID is not equal to your person's user ID" and then switch to text mode and put in additional text mode, like: 

EXISTS:a:$$OBJCODE=ACSRUL

EXISTS:a:securityObjID=FIELD:ID

EXISTS:a:accessorID=your person's user ID

EXISTS:a:coreAction=EDIT

This is a sample exists statement querying access rules for anything where the object ID on the sharing rule is the same as your project ID, and then finding out if your user is on it, and if they have "edit" access (this commonly is what manage access is called). The above is just a guess to point you in a direction, so you might want to explore the API explorer for more tidbits, if it errors. This is all under the Sharing object, which is what Access Rules are commonly referred to in the API explorer.

Edited to add that if the user is in a company, role, group or team that was given manage access, they will likely not be covered with the statement above. So you'll have to discover these separately (you might do this by querying the users company, roles, groups or teams, and then searching to see if any of those objects had manage access to anything.