Hi Antoinne,
To better understanding the situation and to help us reproduce it, could you share more details about this?
1. Did you create the sub-site under the default team site (domain.sharepoint.com)?
2. When you say "3. I checked the permission of sp.user...", may I know whether you checked this via clicking Check Permissions in Site permissions page? If not, what about checking it this way?
3. Is the user in step 4 an internal user or an external user?
About Limited Access, here're some detailed descriptions: The Limited Access permission level is unusual. It enables a user or group to browse to a site page or library to access a specific content item. Typically, the user was given access to a single item in a list or library, but does not have permission to open or edit any other items in the library. The Limited Access permission level includes all the permissions that the user must have to access the required item.
You cannot assign Limited Access permission level directly to a user or group. Instead, you assign appropriate permission to the single item, and then SharePoint automatically assigns Limited Access to other required locations.
For your reference: Default Permission Levels.
Regards,
James