Indian Railways allows dogs in exactly one passenger class: AC First Class (1A). You must book the entire coupe or cabin under one PNR so no other passenger is assigned.
Any train with a First AC coach allows dog travel using this method -- Rajdhani, Duronto, Goa Express, and many long-distance mail and express trains.
The dog box option
Dogs may also travel in the brake van -- a metal enclosure in the luggage compartment. Your dog travels separately from you. For short journeys with a calm crate-trained dog this may be workable. For long overnight journeys it is not something we would choose for Apple.
What about Vande Bharat?
There are two different Vande Bharat trains and the pet rules are completely opposite for each. Most articles get this wrong. Read the correct answer here.