Question

Regarding religion, how were the Mauryan and Gupta empires alike?

Answer 1

First, the notion of 'Hinduism' as a religion is a recently evolved thing to fit into certain norms as criterion for a religion. Discard it if you want to make headway. 
By Chandragupta Maurya's time Vaidicism (Brahmanism) with various shades of ritual purity and cults that don't belong to it, were extant. His mentor 'Chanakya' was a 'pure' Vedic scholar. Jainism in extreme south & west and Buddhism in central, north-central and east of it were predominant and in various proportions with Vaidicism, in rest of country (that included Kabul region after the Seleucid Empire was wound up). Emperor As`oka after becoming Buddhist himself took personal interest in spreading the religion. He sent his sister 'Sanghamitra' with a branch of the sacred Bodhi tree (siting under which Buddha received enlightenment) to Sri Lanka where it is dominant religion; and others to Burma, Thailand and the far east. He got the Buddha's preachings etched into stone in rock edicts all over the country. Buddhism continued to enjoy royal patronage across the country. 
His grandson who was a Budhist too, was killed by his commander-in-chief of the army, 'Pushyamitra S`unga' egged on by Vaidic priests who feared the extinction of their religion from the face of India. This man started his own dynasty but his empire was reduced to a kingdom. After that Buddhism was virtually decimated except for pockets like Andhra region. Under Vaidic onslaught Jainism also lost many adherents. This was the situation at the tine of Christ. 
By seventh century, Vaidicism and the associated cults and rituals bounced back with royal patronage. It was the time of the great 'Hindu' revival (as put by modern historians). This was the time the Great Adi S`ankara (of Kaladi, in Kerala) pionered the resurgence of what is now 'Hinduism' exorcising the practice of animal sacrifice. Broadly it is the shape in which Hinduism is found now. Art & Culture reached pinnacles of glory. The country became 'Hindu' absorbing in it all pre-historic cults including the modern Buddhist remnants as well as the S`aka (Scythian) practices brought in by marauders. It continued past Harsha Empire (when Buddhism saw a brief period of royal patronge), till the Islamic onslaught.

Answer 2

Explanation:

Both the Mauryan and Gupta empires were Hindu. ... Aside from Hinduism, other religions such as Buddhism and Jainism were also part of the cultures of these empires.