Firstly humans are already a kind of animal. Also as a thought exercise, why limit possible genes introduced to just Animalia when plants, fungi and bacteria also have useful genes that could be beneficial to humanity? Conversely animal models of human genetic diseases are already frequently used where genes of interest are replaced with disease causing human variants. There is even a small probability such organisms will occur naturally though horizontal gene transfer (as has been found with bacterial genes in some genomes eg. Bombyx mori). Overall the introduction of novel genes that do not disrupt genes in or around their landing site would mostly be under neutral selective pressure
From the technical point generating such organisms is already standardized with egg/embryo injection being used to introduce the transgenes in mammals (many papers have used mouse/rat, pigs and cattle). Transposons (high risk), integrating viruses (high risk) and homologous recombination (lower risk is CRISPR-Cas is not used) can then be used to integrate the transgenes into the genome for stable inheritance if self replicating exosomes with artificial centromeres are not used.
The main hurdle to generating such organisms are legal and ethical. Introducing transgenes into humans has the potential to avoid congenital and aging related diseases like cancer, increase resistance to pathogens, eliminate malnutrition and improve performance and quality of life but raises issues about equality of access, risks associated with prototyping transgene constructs and uncertainties about insertion sites and other off-target genomic changes caused by transgenesis, short sighted human bias and populism if deciding to replace or remove genes and potential suffering or stigma for subsequent generations. By introducing human genes into animals there are possibilities to make them better models for human diseases and development and compatible for organ transplantation but there is the fear that introducing genes associated with neurodevelopment could make them more capable of perceiving suffering or raise their cognitive abilities to be more similar to humans raising conundrums about legal status and rights. Animals with human genes also could pose a health hazard if it enables them to act as vectors for human diseases. There are also religious and cultural hurdles with beliefs that humanity is somehow fundamentally different from other animals through divine intervention so alterations in either direction would be sacrilege. To avoid such ethical dilemmas many countries have created laws banning the generation of such animals or only grant permission for specific research related purposes.