The Short Version
The claim is largely accurate on its culinary prong: multiple authoritative sources (PubChem/NIH, Britannica, U.S. legal precedent) confirm tomatoes are considered vegetables for culinary, nutritional, and legal/customs purposes. However, the botanical prong is clearly false — tomatoes are botanically classified as fruits (specifically berries), not vegetables. Because the claim uses "and/or," only one prong needs to hold, and the culinary classification is well-established. The inclusion of "botanically" is misleading but does not invalidate the overall statement.