At least in English, pre-Romantic poetry (the so called 'Age of Sensibility') has a similar focus on animals. Northrop Frye claims that a "the curiously intense awareness of the animal world" is one of the defining features of the Age of Sensibility, citing Robert Burns' 'To a Mouse', William Cowper's 'The Snail', and Christopher Smart's 'For I will Consider my Cat Jeoffrey' (he also includes Blake's 'Auguries of Innocence', and though Blake is generally listed as a Romantic, chronologically he fits into the Age of Sensibility as well). Frye attributes this to their sense of '[p]ity without an object', or 'treating everything in nature as though it had human feelings or qualities'. A similar argument has been made (though about the Romantic period itself) by Christine Kenyon-Jones, who claims that there was a gradual shift, starting from around 1500 but reaching its peak in 1800, from viewing animals as 'a part of nature to be subdued for Man's use' to recognising a 'strong fellowship between humankind and other species'. So there were definitely antecedents, even if it did intensify (even Milton had described vegetarianism as the ideal in his 'Elegy VI'). She then connects this to Coleridge's idea that 'everything has a life of it's own', yet we are all 'one Life', meaning that animals are simultaneously connected to and distinct from mankind.
Jones also makes the case that the interest in animals of the second-generation Romantics (Shelley, Byron and Keats) was a way of consciously distinguishing between their view of nature and that of the first generation - Byron and Shelley, she claims, thought of Wordsworth's view of nature as too placid and tame, so their use of wild animals was a way of rejecting this (see Byron's 'Epitaph to a Dog'). There's actually a similar dynamic in Blake's 'The Tyger', where the Tyger's 'Evil' (in the sense of 'the active springing from energy') contradicts, or at least complicates, the idea of natural harmony he presents in 'The Lamb'.
Particularly for Byron and Shelley, their interest in animal suffering (and their vegetarianism) also seems to have been influenced by a preoccupation with and rejection of human sacrifice (see Emily Paterson-Morgan's 'The Bloodless Church') - in Queen Mab Shelley describes the 'demon God' as demanding a 'trade of blood', while Byron compared bull-baiting to the killing of gladiators in the Colosseum (which he denounces in Childe Harold's Pilgrimage). Morgan connects this to their rejection of the Christian idea of necessary blood sacrifice and of the broader notion that a world of suffering and domination could be divinely ordained. John Frank Newton, who introduced Shelley and Byron to vegetarianism, was, at least according to Thomas Love Peacock, a self-described Zoroastrian, who believed that the world of suffering was a creation of a false God, and that the universal adoption of vegetarianism would create a new golden age (though that idea of a golden age of harmony between humanity and animals is also a poetic trope stretching back to Ovid and the book of Isaiah).