[SalesForce] Give Edit access to Partner Community Login Profile user for their Partner

I believe what you are looking for is Super User access:

Super User Access

Users can be assigned super user access to give them access to data owned by other users belonging to the same role or those below in the hierarchy. For example, a Partner Manager with super user access can see data owned by other users in the Partner Manager role and the Partner User roles.

Documentation Reference

When a user has this access they can see data by their peers (same role) and those below them but NOT roles above them.

In order to set this up for a Partner Community:

Go to Communities Settings

• Select the checkbox "Enable Partner Super User Access"
• Save.

Then on the Partner User's contact record:

• Press the "Manage External User" button
• Select "Enable Super User Access". (If this option does not appear, it needs to be added to the Contact's page layout first)

A user on a Customer Community licence can always see the Contact and Account records that they are linked to, regardless of the OWD and Sharing Sets. The Assets being a child with a MD relationship to the Account are visible because the Community user profile has Read and Edit access. The reason that the community user cannot edit the Account or Asset because no sharing set is there to give them Read/Write access on those records.

From the documentation

Access to Records High-volume community users can access the following types of records, based on specific conditions:

• They can access their own account and contact records, based on implicit sharing.

• They have “Read” access on the account they belong to.

• They can access a record’s parent, and the organization-wide sharing setting for that record is Controlled by Parent.
• The organization-wide sharing setting for the object is Public Read Only or Public Read/Write.