The deity of Purusha Suktam is none other than Shri Hari Vishnu. We shall see what the scriptures say and also see why some other deities are praised with Purusha Sukta.
नमस्ते वासुदेवाय नमः सङ्कर्षणाय च ।
प्रद्युम्नायनिरुद्धाय सात्वतां पतये नमः ॥
Obeisances to You, Lord of the Sātvatas, and to Your forms of Vāsudeva, Saṅkarṣaṇa, Pradyumna and Aniruddha.
First, the Sruti itself explains the meaning of Purusha Sukta and also clears doubts regarding the identity of Purusha Sukta's deity. The first Khanda of the Mudgala Upanishad says
Hence the first three Mantras of Purusha Sukta talk about the Sarvavyapakatva(omnipresence), Sarvajnatva(omniscience) [Regarding him being everything that existed, exists and will exist] and Sarvasaktimatva(omnipotence) [He having the characteristic of giving Moksha]. Hence the Mudgala Upanishad's explanation of Purusha Sukta establishes Narayana as the Param Brahma and says that the next mantra talks about his Chaturvyuhas(namely Vasudeva, Sankarshana, Pradyumna and Aniruddha).
The last chapter of the Mudgala Upanishad says
Hence the correct explanation of Purusha Sukta must not be preached to the ones who are not Vaishnavas.
Now, what about the Puranic statements of Purusha Sukta being used in Shiva's worship and Shiva's matching descriptions with the Purusha?
This can be explained as well. The Linga Purana says
प्रत्येकमश्वमेधस्य यज्ञस्य सममुच्यते ।
य एकः पुरुषः श्रेष्ठः परमात्मा जनार्दनः ।। (Linga Purana 2.1.7)
The remembrance of Nārāyaṇa, his worship, the devoted obeisance to him—every one of these holy rites is on a par with the horse-sacrifice. Janārdana alone is that excellent Puruṣa who is the Paramātma.
Hence Vishnu alone is the Maha Purusha. We have an explanation regarding Shiva and even Brahma sometimes being attributed the characteristics of the supreme purusha.
In the same manner in which the many Purushas are said to have one origin in the Supreme Purusha, it may be said that this entire universe is identical with that one Purusha of superior attributes. I shall explain this now, after bowing to my preceptor Vyasa, that foremost of Rishis, who is conversant with the soul, endued with penances, self-restrained, and worthy of reverent worship. The Purusha Sukta, O king, occurs in all the Vedas. It is well known to be identical with Rita and Truth.
-Mahabharata, Santi Parva, Chapter 351
Hence many Purushas are born from the Maha Purusha and that's why they can be identical with him as they have him as their soul. Now who is this Maha Purusha? Brahma explains this to Shiva in the next chapter.
Some persons possessed of learning speak of him as the supreme soul. Others regarded him as the one soul. A third class of learned men describe him as the soul. The truth is that he who is the Supreme Soul is always divested of attributes. He is Narayana. He is the universal soul, and he is the one Purusha. He is never affected by the fruits of acts even as the leaf of the lotus is never drenched by the water one may throw upon it.
-Mahabharata, Santi Parva, Chapter 352
Hence wherever the Purusha Sukta is being attributed to other Devatas, it doesn't mean that the Sukta belongs to them. We can come across many places where Shiva, Brahma, Surya and even Devi Lalita were worshipped with this Sukta. But as explained by Mahabharata it belongs to the Maha Purusha who is none other Hari who is the inner soul of everyone.
Otherwise it contradicts Sruti that strictly restricts the preaching of the Purusha Sukta's meaning to Non-Vaishnavas.