It's a genuinely common assumption that Eurostar works the same way as Eurotunnel Le Shuttle for pets, and it doesn't. Eurostar does not carry pets on any route to or from London, nor on routes between Brussels or the Netherlands and Lille, nor on its seasonal Sun and Snow services. The only exception on these routes is assistance dogs, and even then Eurostar asks to be contacted at least 24 hours before travel.
On other mainland European routes — travel between Belgium, France, Germany and the Netherlands that doesn't touch London — Eurostar does allow pets. Small animals under 6kg travel free in a compliant carrier no larger than 45 x 30 x 25cm. Larger dogs that don't fit in a carrier need a dog fare, bought from the Train Manager on board, and must stay leashed, muzzled, and either on your lap or the floor of the coach for the whole journey.
The practical takeaway for a UK-to-France trip: if you were planning to bring a pet on the train under the Channel, it has to be Eurotunnel Le Shuttle, not Eurostar. Eurostar only becomes an option for a pet once your journey is entirely within mainland Europe and doesn't touch a London route.