In short term, the layout assignment information is listed in RecordType instead of Profile.
When you retrieve whole SObject Describe from this endpoint:
/services/data/v40.0/sobjects/Account/Describe
The response have a "recordTypeInfos" section like this:
[
{available: true, defaultRecordTypeMapping: true, master: false, name: "Account Name 1", recordTypeId: "xxx", …},
{available: true, defaultRecordTypeMapping: false, master: false, name: "Account Name 2", recordTypeId: "xxx", …},
{available: true, defaultRecordTypeMapping: false, master: true, name: "Account Name 3", recordTypeId: "xxx", …}
]
For each recordTypeInfo, it contains a value called url:
url:{layout:"/services/data/v40.0/sobjects/Account/describe/layouts/YourRecordTypeId"}
If you send a GET request to this endpoint, in the response you will see the Layout Id in:
{id: YourLayoutId}
So the mapping is already done by salesforce. That layout id reflects the layout assigned to the current user.