A hobby business is one for which there is no "profit motive." The only difference between a hobby business and any other business is that a person without a profit motive is not liable for self-employment tax on their earnings. Instead, they report their income as Miscellaneous, rather than on a Schedule C.
If there is a significant profit to a hobby business, the IRS will likely challenge the taxpayer's position, so, if your daughter is not in the mood for an audit, then taking this position on her tax return may not be the best move. But, if the income doesn't do much more than defray the cost of the animal, then there probably will never be an issue, because the IRS is not interested in small issues like this.
That said, a DBA is not necessary, but it will help identify the business. A taxpayer ID is necessary if your daughter wants to keep transactions in a separate bank account from her other person activities.
Stud services are tax exempt, however form ST-125 must be filed. Here is a guide to NY State Sales and Use Tax.
Hope this helps.
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