If all associated attributes of an intended primary key are identical, is it still a true primary key?

I'll start with an example. If I have a person table with intended surrogate primary key Id:

+----+------+------------+-------------+| Id | name |    DoB     |     SSN     |+----+------+------------+-------------+|  1 | John | 1901-01-01 | 111-11-1111 ||  2 | Jane | 1902-02-02 | 222-22-2222 ||  3 | John | 1901-01-01 | 111-11-1111 |+----+------+------------+-------------+

Note Ids 1 & 3 have the same attributes; They both represent the same person.

Now from what we know about the theory behind what constitutes a primary key, which I think is well summarized here:

• The primary key must uniquely identify each record.
• A record’s primary-key value can’t be null.
• The primary key-value must exist when the record is created.
• The primary key must remain stable—you can’t change the primary-key field(s).
• The primary key must be compact and contain the fewest possible attributes.

Consider the first bullet, "The primary key must uniquely identify each record." In my example, I suppose whether or not each Id does represent uniqueness depends on what's really supposed to be considered unique. A different database record? Yes. A different person (what the records are supposed to represent)? No.

So multiple Ids represent what is functionally the same subject generating the data, present in 2 records. A "two to one Id" of sorts. I've not read anything that directly address the scenario my example illustrates, as is relates to what is or is not a PK.

1. Does this example violate the theory behind what constitutes a primary key?
   2. If not, does this example illustrate a violation of any larger principles of database architecture, or can this concept be reduced to something as simple as "duplication of data - clean it up"?

Many thanks.