Is the title 'Vishnu Sahasranama' a later addition or originally attributed to Lord Vishnu? The question arises because the names Brahma, Ishwar, Siva, Durga etc also find mention in the Sahasranama.
Vishnu sahasranama refers to 1000 names of Lord Vishnu , who is infact the Para Brahman. Let us see the nature of the question raised which leads to revelation Vishnu sahasranama by Bheeshma.
Yudhisthira raises real fundamental questions.
Kimekam daivatham lokay kim vapyekam parayanam !
Sthuvantham kam kamarchanthaha prapnuyurmanavaha shubham!!
Ko dharmaha sarva dharmanam bhavathaha paramo mathaha !
Kim japan muchayatey janthurjanmasamsara bandhanath!!
Yudhisthira is basically asking questions which means
" Who is the supreme Lord who when worshipped provides provides succor, auspiciousness etc and also the ultimate deliverance from cycle of life and death. Who is such ultimate supreme?"
No where yudhisthira has mentioned any deity. To this Bheeshma says it is Lord Vishnu who provides moksha.
Bheeshma then goes on to describe the thousand names of the supreme lord which is famous as Vishnu sahasranama.
Such a background is not there for sahasranamas of other deities, in the sense like the unique and generic questions raised by Yudhisthira.
So, Vishnu Sahasranama alone describes the ultimate Para brahma, who is known as Lord Vishnu\Narayana.
Because, Vishnu sahasranama talks about the original Supreme Lord , so all names like sambhu, siva, sthanu etc also apply to Lord Vishnu only. Moreover, the names like Siva, Sambhu, Isana, Sthanu are all common nouns which are very apt for Lord Vishnu only.
In introducing the Vishnu Sahasranamam, Bhishma specifically says that he is going to recite the thousand names of Vishnu:
tasya lokapradhānasya jagan nāthasya bhūpate
viṣṇor nāma sahasraṃ me śṛṇu pāpabhayāpaham
Hear, O king, the thousand names, possessed of great efficacy in destroying sins, of that foremost one in all the worlds that Master of the universe, viz., Vishnu.
He makes a similar statement after he finishes reciting it:
Even thus have I recited to thee, without any exception, the thousand excellent names of the high-souled Kesava whose glory should always be sung.... This hymn in praise of the illustrious Vishnu composed by Vyasa, should be recited by that person who wishes to acquire happiness and that which is the highest benefit (viz., emancipation).
As far as whether its original name is the Vishnu Sahasranamam, not exactly. As opposed to the rest of the Mahabharata, the Vishnu Sahasranmam consists of mantras, not shlokas, and thus it is said with a preamble, which says this:
asya sri viśṇūr divya sahasranāma stotra mahā mantrasya|
sri vedavyāso bhagavān Rśiḥ|
sri mahāviśṇu paramātmā sriman nārāyaṇo devatā|
Of the great mantra called the Sri Vishnor Divya Sahasranama Stotra, the sage [who heard it from the gods] is Sri Vedavyasa Bhagavan, the meter is Anushtup, the deity [whom it is about] is Sri Mahavishnu Paramatma Sriman Narayana.
So it's technically known as the Sri Vishnu Divya Sahasranama Stotra.
As to why names of other gods are included, in some cases it might seem like it's the proper name of some familiar god, but it's actually being used with its literal meaning. So for instance, the name Ishwara is often used as a personal name of Shiva, but literally it just means "the lord" and is thus applicable to Vishnu. In other cases, names of other gods are used intentionally to denote that as Paramatma (the supreme soul), Vishnu dwells as the Antaryami of all the gods. As the Vishnu Purana says, "The term Vásudeva means that all beings abide in that supreme being, and that he abides in all beings."
By the way, there are also Sahasranamams for other gods. In fact the Anushasana Parva of the Mahabharata, which contains the Vishnu Sahasranamam, also contains a Shiva Sahasranamam, as I discuss in this question. (It's not recited by Bhishma though.)