I am not an expert on Mahayana Buddhism, but I will try to answer this question.
I found another translation here.
For the same section quoted by OP, this other translation states:
Bhikṣus, thus you should not cultivate your ideas of impermanence,
suffering, nonself, impurity, and so forth as if they were the true
meaning [of the human condition]. Just like those people who took
rocks, vegetation, or gravel to be the jeweled necklace they were
looking for, you must carefully learn about expedient means so that
wherever you are, you can continually cultivate your perception of
self as well as your perceptions of permanence, bliss, and purity.
Later on, the text says:
And out of my desire to subdue the non-Buddhist paths I therefore
declared: “There is no self,there is no person or [individual] living
being, life span, personality, observer, actor, or experiencer.”
Monks, the heterodox paths affirm a “self” in the same manner as [some
infer literacy in] the shapes of letters incised into wood
accidentally by insects. This is why the Tathāgata proclaims “nonself”
as part of his buddha-dharma. It is because I need to straighten out
[the thinking] of living beings—because I am aware of their
situation—that I expound the absence of self. ....
But what I am speaking of is not what ordinary people imagine the self
to be. Ordinary people or ignorant people suppose the self to be the
size of a thumb, or perhaps a mustard seed, or a speck of dust. What
the Tathāgata explains the self to be is nothing like that. Therefore
when I preach “dharmas are without self,” in truth they are not
without self. So what is this self [of which I now speak]? If a dharma
is true, real, permanent, autonomous, a basis, and its nature is
immutable, then that is what I call self.
The term "dharma" above means "thing". So "self" is a thing which is true, real, permanent, autonomous, a basis and is immutable.
Then it continues:
[Kāśyapa continued:] “World-Honored One, is there a self or not in any
of the twenty-five forms of existence?”
The Buddha said: Good man, “self” is precisely what tathāgatagarbha
means. All living beings have buddha-nature, and this is what is meant
by this notion of self. However, the significance of “self” understood
in this way has been continuously covered over by an uncountable
number of the defilements since the beginning [of any given
individual’s existence], and that is why living beings have been
unable to perceive it.
So, the "self" in the Mahayana Mahaparinirvana Sutra refers to the tathagatha-garbha or Buddha-nature.
It must be noted that tathagatha-garbha or Buddha-nature is not the same as the concept of Atman in Hinduism, as can be seen by the following excerpt from the same text Mahayana Mahaparinirvana Sutra:
Good man, if people have never heard of the profound, hidden tathā-
gata treasury, how could they be aware of the existence of
buddha-nature? What is it that I am calling “a hidden treasury”? I am
talking about the well-balanced Mahāyāna scriptures themselves. Good
man, there are other paths: some expound a self that is permanent,
some expound a self that is cut off permanently [at death]. The
Tathāgata is not like them. In affirming both self and nonself, I call
it “the middle path.” One way of explaining this is that the Buddha
expounds a middle path in which all living beings possess
buddha-nature, but because it is obscured by the defilements they do
not understand it and do not see it. Therefore you must diligently
cultivate whatever expedient means you can in order to destroy those
defilements.
According to this article:
Tathagatagarbha, or Tathagata-garbha, means "womb" (garbha) of Buddha
(Tathagata). This refers to a Mahayana Buddhist doctrine that Buddha
Nature is within all beings. Because this is so, all beings may
realize enlightenment. Tathagatagarbha often is described as a seed,
embryo or potentiality within each individual to be developed.
And it discusses further:
In the religions of the Buddha's day that were the forerunners of
today's Hinduism, one of the central beliefs as (and is) the doctrine
of atman. Atman means "breath" or "spirit," and it refers to a soul or
individual essence of self. Another is the teaching of Brahman, which
is understood as something like the absolute reality or the ground of
being. In the several traditions of Hinduism, the precise relationship
of atman to Brahman varies, but they could be understood as the small,
individual self and the big, universal self.
However, the Buddha specifically rejected this teaching. The doctrine
of anatman, which he articulated many times, is a direct refutation of
atman.
Through the centuries many have accused the Tathagatagarbha doctrine
of being an attempt to sneak an atman back into Buddhism by another
name.
In this case, the potentiality or Buddha-seed within each being is
compared to atman, and Buddha Nature -- which is sometimes identified
with the dharmakaya -- is compared to Brahman.
You can find many Buddhist teachers speaking of small mind and big
mind, or small self and big self. What they mean may not be exactly
like the atman and Brahman of Vedanta, but it's common for people to
understand them that way. Understanding Tathagatagarbha this way,
however, would violate basic Buddhist teaching.
Another article here tries to point the origins of tathagatha-garbha or Buddha-nature to the Luminous Mind in the Pali Canon (which according to Theravada Buddhism, is not permanent or eternal, but is dependently arising - see this answer):
The term "'tathagatagarbha'" is generally taken as to mean that the
"garbha" (womb or potential) of a 'Tathagata' exists in all sentient beings without
exception, and though temporarily contaminated by adventitious
defilement ('agantukaklesa'), it is the cause which eventually leads
sentient beings to enlightenment. The notion of the 'tathagatagarbha'
can be traced to a luminous¡A inherently pure mind (pabhassar citta)
found in the 'Anguttara-nikaya' (1:5):
Oh! 'Bhiksus'. The mind is pure! It is defiled by the adventitious
defilement. Oh! 'Bhiksus'. The mind is pure! it obtains liberation
through the adventitious defilement.
When the original pure mind came to be regarded as something capable
of growing into Buddhahood, there was the 'tathagatagarbha' doctrine.
Although the concept of an intrinsically pure mind exists in the
Nikaya Buddhism, many Buddhologists, such as Wayman (1), Paul (2),
Yin-shun (3) think that the 'tathagatagarbha' thought was originated
from the 'Mahasamgika', but was rejected by the 'Theravada'. This
theory is also held by Mizuno who points out that the pure mind
('pabhassarcitta') articulated in the Nikaya Buddhism is not totally
identical with the original pure mind ('prakrtivisuddhi-citta')
articulated in the 'Tathagatagarbha' doctrine, for Mizuno asserts that
the former is static whereas the latter is dynamic in that it is
capable of eradicating defilement.(4) At any rate, the relationship
between pure mind and the adventitious defilement appears to have been
wholly adopted by the 'Mahasamghika' and later by the 'Mahayana'.
It also comments in its conclusion:
In conclusion, when we try to interpret the thought of the
'tathagatagarbha', we should keep several points in mind:
1) The 'tathagatagarbha' symbolizes the potential for enlightenment (a
principle) rather than a material "essence" of ultimate truth,
2) the 'tathagatagarbha' is based on the framework of the 'Mahayana'
doctrine of 'sunyata-pratitya-samutpada'.
3) The development of the 'tathagatagarbha' doctrine signifies the
ability of a religious tradition to meet the spiritual needs of the
masses aiming at a given time.
That is to say the 'tathagatagarbha' thought was formed as an positive
soterio-logical approach to counteract the "'sunyam sarvam'" (all is
empty) view. The 'tathagatagarbha' which strongly articulates a
devotional and experiential approach to salvation provides much to the
hope and aspiration of the people at large. It is this positive aspect
that was taken up and strongly emphasized in Chinese Buddhism.
4) The 'tathagatagarbha' doctrine is employed as a skill-in-means
('upaya'). This does not necessarily mean that the theory of the
'tathagatagarbha' is neyartha, a teaching requiring further
qualifications -- rather, it is a skill-in-means in that it is taught
to suit the needs of a certain kind of people and circumstances. This
is why it is said in the 'sutra' that in order to teach the emptiness
of all dharmas, the Buddhas preach sometimes by the doctrine of the
'tathagatagarbha', and sometimes by that of emptiness. Thus it is
better to take the 'tathagatagarbha / Buddha nature' as representing
"profound existence" derived from "true emptiness" rather than as a
monistic self.