Yes. Hinduism allows the worship of sages and saints. In fact, we deify gurus and not just worship them, we worship their slippers (pAdukAs). Examples of sages and saints worshipped in temples:-
Mantralaya Raghavendra Swami (madhwa saint and guru)
Nrsimha Saraswati
Akalkot Maharaj
Manikya Prabhu and others in the Dattatreya Sampradaya.
We worship at the shrines of previous acharyas in the Sankara mathas.
We worship the sankara pAdukAs in Sankara mathas.
Sankaracharya himself
Saint SantRam in Nadiad
Bapa Sitaram in Kathiyavad, Gujarat
Shirdi Baba was worshipped in Hindu tradition even while he was alive. After he left his mortal coil, the worship continued in the same fashion. I suspect one of the reasons that Shirdi Baba is worshipped in Hindu temples is due to the government's Endowments Department which noticed Babaji's following and seized the business opportunity. They started placing navagrahas and other idols in Babaji's temples so that they can claim the temple's earnings. So, basically Hindus took over Baba and not the other way around. I do not have sources to back this up except hearsay.
Personal note: I am an ardent devotee of Shirdi Sai Baba.
why other Hindu saints are not worshipped
Well, at least in Maharashtra there are many examples of saint who are worshipped like God, for example: Sant Gajanan Maharaj, Sant Gyaneshwar, Sant Tukaram. Their birthdays are also celebrated with Aarti and other rituals. So, Sai baba are not unique in that sence