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First of all - "Govinda" not being "Narayana" can refer to any deity even by Vaishnavite rules.

But then a blog named [shastranethralaya](
http://blog.shastranethralaya.org/?p=165)

says

Normally this is taken as a Bhakti text. But we’ll see it in its true form, like any other Acharya’s text this is also another which deals only with Vedanta.

So this stanzas summary is, they have to worship the Lord who is none other than the Self instead of wasting our time in different forms of action. Since no action can make us attain the ever attained.

Bhaja Govindam stotra is attributed to Krishna or Vishnu.

The author of Bhaja Govindam stotra is Adi Shankaracharya. He also wrote commentary on Vishnu Sahasranama. He gave a detailed explanation of the name Govinda. Not one or two, more than that.

From The Vishnu Sahasranama Bhashya with Adi Shankara commentary"

The Mahabharata, Mokshadharma (San., 343-41) says: "I first knew (vinda) the Earth (go) which was carried away and hidden in a cave (by an Asura), hence I am praised by the appellation 'Govinda' by Gods, and Scriptures." The Harivamsa (76-45 and 279-49) says: "I am Indra (Lord) over the Devas, thou art lord over the cows, hence the world will ever praise Thee as , Govinda '; speech is named 'Go,' and since thou conferest speech, 0 Lord, the Sages call you 'Govinda'."

Ka Go+ Vide has the following tell meanings: Go

1. Svarga (Heaven). He transcends the Heaven.

2. Arrows (He knows all the weapons).

3. Cattle. He is the leader of the ignorant ones.

4. Speech. He is to be known by the Vedas.

5. Thunderbolt; He has the Vajra marks on his feet.

6. Quarters; He is known in all quarters.

7. eyes; He is in the person residing in the eyes.

8. The sun ; He is in the form of the Sun.

9. Earth; He recovered the Earth from the Titans.

10. Waters; His seat is in the waters.

And Vid, to know.

He quotes the incident of Lord Vishnu's avatar Varaha saving Earth from Hiranyaksha and also Harivamsa incident where Indra praises Lord Krishna to show the name Govinda referring to Vishnu and Krishna (Adi Shankaracharya does not see difference between the two). Hence, the author's words need to taken as a reference and authority.

Hence, we can undoubtedly say Bhaja Govindam stotra is dedicated to Lord Krishna.

It is a Bhakti text because it says the people to recite the name of God. It recommends devotional path by chanting Lord's name. It is also a Vedanta text because in the stotra, Adi Shankara explains how a person in a samsara totally wastes his time assuming wealth, family and mortal life etc., as important. He asks us to concentrate on the reality and get out of the maya. It was originally called Moha Mudgaara (a hammer to shatter illusion). Vedanta teaches that there is only one reality and asks us to get out of maya. Hence, the stotra is not only a Bhakti text but also a Vedantic text.

Original question title before OP edited it:
Does "Govinda" in the "Bhaja Govindam" prayer attributed to Adi Sankara refer to Krishna ?

From the VERY EXACT SAME blog article, at the beginning of the shloka's explanation

Bhaja Govindam Bhaja Govindam – Definetly worship Govinda (Krishna)

I have bolded the word Krishna (it's in black color) in case it's not easily visible.

It may be treated as a Bhakti text - but its deeper meaning is Advaitic.

https://advaitavedaanta.wordpress.com/2015/04/21/bhaja-govindam-verse-1/

भजगोविन्दं भजगोविन्दं गोविन्दं भजमूढमते |

संप्राप्ते सन्निहिते काले नहि नहि रक्षति डुकृञ्करणे ||१||

bhaja govindam bhaja govindam, govindam bhaja mūḍḥamate

samprāpte sannihite kāle nahi nahi rakṣatiḍukṛnkaraṇe – 1

What should I do? bhaja govindam bhaja govindam govindam bhaja mūḍḥamate. bhaja means worship. bhaja govindam means worship Govinda. What is Govinda? Govinda is that freedom, the wholeness, the immortality, and the totality that I am seeking. In common usage, Govinda means Lord Kṛṣṇa. The word go means cow. The one who protects the go is Govinda, the Lord was a cowherd, so he was called Govinda. But the word go has several other meanings as well. When we analyze the meanings of the word Govinda, we understand that, in this context, Govindameans the limitless Self.

i. Another meaning of the word go is prtvī or earth. Lord Kṛṣṇa is called Govinda, the protector of the earth.

ii. go can also be taken to mean the sense organs. Govinda, then, is the one who impels the sense organs, the one who enlivens the sense organs, or the one who directs the sense organs to perform their appointed functions.

iii. The word go also means speech. In this sense, it means the one who is behind all speech. Govinda is the one because of whom all speech occurs, and because of whom words have the ability to reveal their meanings.

iv. Thus, the name Govinda is not necessarily limited to Lord Kṛṣṇa. In a broad sense, it means the Self, the truth or the Consciousness. It is the happiness that is behind every name and form and, indeed, that which is behind everything in the creation. That Self is Govinda.

भज् has lots of meanings:

revere, worship, adore

which make sense in the Bhakti interpretation

or

to apply one's self to, pursue, practise, cultivate, be engaged in;

which is the deeper intended meaning.

"Worship Krishna" is the meaning for ordinary folk and "be engaged in the Atman" is the meaning for Advaitic seekers.

Yes, it seems so.

Because, bhajanaa comes from the root "भज्" which commonly means worshipping. It also has several other meanings. please check: http://sanskritdictionary.com/?iencoding=iast&q=भज्&lang=sans&action=Search

As Swami Vivekananda points out

"নির্গুণ পরব্রহ্মের এই নির্গুণ স্বরূপ অতি সূক্ষ্ম বলিয়া প্রেম বা উপাসনার পাত্র হইতে পারেন না" meaning that the Nirguna Swarupa of Para-Brahman can not be worshipped as the form is very very subtle (quoted in Sri Sri Totapuri-Prasanga, Publisher: Netai Chandra Basu, Lake Kalibari, ISBN 978-81-935349-1-5).

According to Sri Trailanga Swami, Nirguna Brahman as The Supreme Self can not be worshipped. It has

to be known by Neti Neti Vichaara in Jnanamarga (Reference:Mahavakyaratnabali, Sri Sri Trailanga Swami, quoted in Ibid page 77).

These opinions I value morevas they are coming from realised saints. Otherwise, there IS one Brahma-Stotra, commonly known as written by Sankaracharyya, which reads

Tadekam smaramah tadekam bhajaamah where bhajana is being used for a separate meaning.

, But we have to remember that this advice is meant for common masses.The stotra repeatedly uses the expressions like

sakridapi yasya na Murari-samarchha, tasya yamah kim kurute archaam

iha samsaare khalu dustaare kripaaya apaare paahi Muraare

gyeam Gita-nnamasahasram ghyeam Sripati-rupam ajasram

(Reference:Stavakusumanjali, Udbodhan, page 388-89)

In the same stotra, different names like Murari, Sripati and Govinda are being used. Murari and Sripati means Sri Krishna or Vishnu quite definitely.

And that the stotra is meant for the nasses is clear from its describing the different stages of life like youth and old age and their limitations.(ibid oage 309-393).

For following the Jnana-Marga for knowing the Supreme Self, six properties sama,dama, titiksha, uparati,sraddha, samadhana are neede alongwith mumukhutwa.Ibid page 72.

So here I think Govinda means Sri Krishna and not the Formless Para-brahman.

All do not believe that this stotra is composed by Adi Shankaracharyya by the way.