Not all lingayats are demanding Lingayatism as separate religion. Only a few lingayat are demanding and that too due to political reasons.
Let us analyze their supposed philosophical differences:
1) Those who demand Ligayat as separate religion, they say Lingayats accept Shiva which is formless, which is different from other Hindus.
Answer: As per scripture: Now, shiva is formless, but Shankara is the form which dwells in mount kailasha śivaiko brahmarupatvānniṣkalaḥ parikīrtitaḥ [Shv. Pu. - 1.5.10]
- Shiva alone, being Brahman, is known as formless and quality less.
in the Gita Shri Krishna says He is Shankara among all the rudras: rudrāṇāṁ śaṅkaraś cāsmi [BG - 10.23] - Of all the Rudras, I am Shankara So it means Shankara is a Rudra. Hence, although Shiva and Shankara both imply the same deity, Shiva means his formless pure bright light form (meditated upon by Yogis) and Shankara means his commonly known Kailasa dwelling ascetic form (worshiped by devotees). Also see: Are Lord Shiva and Lord Shankar two different deities?
--> Hence, Lingayat accept Shiva, which is accepted by other sects, but not accept Shankara. So it is not a big difference. Because other sects already accept formless Shiva. Apart from that, Arya samaj of Hinduism accepts "only" formless Ishwar.
2) Supporters of Lingayat as separate religion say that "Lingayats do not beleive in Puranas and any mythological stories about Shiva or any other God"
Answer: Arya Samaj also rejects Purana or any other shashtra as non-authoritative.
3) Supporters of Ligayat as separate religion say that "Ligayats do not beleive in caste"
Answer: Arya samaj, Natha Samparadaya, and innumerable sects of hinduism do not beleive in caste.
4) Supporters of Ligayat as separate religion say that "Lingayat only worship Isht Linga"
Answer: Veerashaiva also worship Isht Linga http://www.shivayoga.net/ishta-linga-2/what-is-ishta-linga/ http://www.shivayoga.net/veerashaivism/
Arya samaj(part of Hinduism) do not even beleive in Linga or any other object or idol.
5) Supporters of Ligayat as separate religion say that "Basavanna rejects Vedas"
Answer: please see https://hinduism.stackexchange.com/a/23124/13287
At one place Basavanna has used vedas as authoritative/proof. Hence it shows Basavanna was using vedas as proof.
Do not say, out of conceit, that the gods number two or three For He is but one, and to say there are two is a lie The Veda avers: there is none but Kudalasangamadeva
Shall I say that the Sastra is supreme? It glorifies karma! Shall I call the Veda great? It talks about animal sacrifice! Shall I call Smriti great? It is turned towards the future! Since you abide nowhere there Only in the three modes of services You are present Kudalasangamadeva. ref: https://books.google.co.in/books/content?id=kITXAAAAMAAJ&pg=PA211&img=1&pgis=1&dq=%22Since+you+abide+nowhere+there+Only+in%22&sig=ACfU3U2AXarng8LyOyJbZJ4nIzxZh0ZA-w&edge=0
Hence, in this verse he has criticised animal sacrifice part in vedas
Now, here arya samaj also has objected to translations where animal sacrifice have been alleged in vedas. http://www.thearyasamaj.org/articles?=160_Animal_Sacrifice_Before_Deities Arya samaj rejects animal sacrifice interpretation of vedas. Vaishnavas also reject animal sacrifices
I shall sheathe the Vedas, fetter the Sastras, Put Tarka in tongs, deface the Agamas. Look father, most generous Kudalasangamadeva, I am a son of Madara Chennayy’s family. ref: http://lingayatreligion.com/Equality_of_Basava.htm
Hence, in this verse Basava has just said that put vedas in sheathe.
Now, inside other sects in hinduism:
- Sant Mat,Kapalika,Sri Vidya have their own scriptures which they value, and they do not hold Vedas as canonical authority.
- Mimamsa argued that the Vedas could not have been authored by a deity
- Even In Gita, Krishna referred that in certain aspect like Moksha, Vedas don't hold supremacy. e.g. BG 2.45.. refer: Do any Hindu scriptures say that Hinduism is not a religion but simply a way of life?
6) Supporters of Ligayat as separate religion say that "Lingayat is monotheist, whereas Hindus are polytheist"
Answer: Very few Hindu sects and non-denominational are polytheists, Many of them are Henotheist. Viashnavaism, Shaivism etc are Monotheist. Arya samaj is even more staunch monotheist
rather at https://books.google.co.in/books/content?id=kITXAAAAMAAJ&pg=PA204&img=1&pgis=1&dq=indra+amritavati&sig=ACfU3U1uJPic-9wS0VzUl-c03mKNxjeodw&edge=0 Basava says the following in an attempt to prove that Shiva is supreme:
Indra was born to Amritavati and Somasambhu
Brahma was bom to Satyarishi and Jyesthadevi
Visnu born to Vasudeva and Devaki
The Enlightened was born to Nabhiraj and Marutadevi
All known to the worlds as fashioned out of the womb
Has Lord Kudalasangamadeva, who lives outside
The cycle of birth and death
Any father and mother?
That is, here Basava is still accepting Indra etc.
- but Arya samaj(part of Hinduism) even rejects Indra etc.
- Poorva mimansa does not accept any independent existence of God other than mantra itself
Hence, it proves that all the proposed logic of separate religion are invalid; and it is just a political game being played time to time by few lingayats.