Kolkata has a genuine dog cafe culture spread across several neighbourhoods, and every cafe on this list is Team Verified with no pet entry fee at any of them.
Barks & Brew in Kalighat bills itself as the city's first pet cafe and store, with a separate home-cooked pet menu. Cup eBong Cafe in Hindustan Park and Mrs. Wilson's Cafe in Jodhpur Park both welcome dogs and cats with no charges, the latter known for its tree-lined balcony seating. Craft Coffee Experience Centre in Salt Lake has no size or breed restrictions and free parking on site. The Backyard in East Topsia is the one exception worth knowing, it requires a leash and a vaccination card at entry, a reasonable ask but worth being prepared for.
Most allow dogs both indoors and in outdoor seating, so weather generally isn't a factor in choosing between them. If you're after a full sit-down meal rather than coffee and cake, it's worth knowing several of these double as pet friendly restaurants in Kolkata too, not just cafes, Blue Bridge Restro Cafe in particular runs a full continental and Chinese menu.