Approach:

All wanted tags are siblings of each other. So, basically, you'll have to use find_next_sibling() function to find them.

Explanation:

All the names (titles) of bird types are located inside the <h2> tag. But, the first <h2> tag is for Contents (so skip that). Order and Family are located inside the <p> tag which comes after the <h2> tag. You can find that using h2.find_next_sibling('p'). The table with Name, Binomial, and Status can be found using h2.find_next_sibling('table').

Using all this, you can print all the details you want. But, you'll have to break the loop when you reach the <h2> tag which contains References. This can be done using

if h2.find('span', class_='mw-headline').text == 'References':
    break

Code:

r = requests.get('https://en.wikipedia.org/wiki/List_of_birds_of_Trinidad_and_Tobago')
soup = BeautifulSoup(r.text, 'lxml')

for bird in soup.find_all('h2')[1:]:
    title = bird.find('span', class_='mw-headline').text
    if title == 'References':
        break
    print(title)
    p_tag = bird.find_next_sibling('p')
    order, family = [x.text for x in p_tag.find_all('a')]
    table = p_tag.find_next_sibling('table')
    for row in table.find_all('tr')[1:]:
        name, binomial, status = [x.text for x in row.find_all('td')]
        print(order, family, name, binomial, status, sep=' | ')
    print()

Partial output:

Tinamous
Tinamiformes | Tinamidae | Little tinamou | Crypturellus soui | Trinidad only

Screamers
Anseriformes | Anhimidae | Horned screamer | Anhima cornuta | Trinidad only - rare/accidental

Ducks, geese, and waterfowl
Anseriformes | Anatidae | Fulvous whistling-duck | Dendrocygna bicolor | Trinidad only
Anseriformes | Anatidae | White-faced whistling-duck | Dendrocygna viduata | Trinidad only - rare/accidental
...

...
Waxbills and allies
Passeriformes | Estrildidae | Common waxbill | Estrilda astrild | Trinidad, accidental Tobago - introduced species
Passeriformes | Estrildidae | Tricolored munia | Lonchura malacca | Trinidad only - introduced species

Old World sparrows
Passeriformes | Passeridae | House sparrow | Passer domesticus | Trinidad only - Introduced species