Safety is mostly about the dog in front of the bowl. Omega oils are generally well tolerated, but higher-fat additions can trigger loose stool, vomiting, or flare-ups in dogs who do poorly with rich foods. Controlled feeding research in dogs has evaluated safety of EPA/DHA sources in diet contexts, supporting that these ingredients can be used safely when amounts and the overall diet are appropriate (Dahms, 2019).
At home, the earliest warning signs are greasy diarrhea, repeated lip-licking after meals, or a dog that suddenly skips breakfast when oil is added. If those appear, stop the new addition and contact the veterinarian rather than pushing through. Also remember the calorie piece: if oil is added daily, some dogs need a small food adjustment to avoid slow weight gain.