No, not every family would have owned a Bible (many people in early 1800s especially would have been illiterate) and not every family Bible contains family details. There's no firm rule as to what could be written inside a Bible- sometimes it's just names and dates of births, marriages and death and other times there are loads of details included.
My great-great-great-grandfather owned a beautiful Bible which was given to me about 20 years ago. His signature appears with lovely flourishes at the start but the printed pages for recording family events are completely blank.
Another old Bible in our house has wonderful information inside such as "Thomas Armstrong departed this life July 3d 1846... I placed his son Thos. with Morgan ___ in July 1849- to lodge and feed him- he is now 4 years and four months old. June 15th 1850. Jane Phibbs
It is my intention with the blessing of my heavenly Father to educate him in the protestant faith, and if spared to apprentice him at a proper age to a trade.- Jane Phibbs- June 15th 1850."
And not all 'Family Bibles' are old- we also have one that my in-laws got as a wedding present.
There can also be 'bits' of paper tucked between the pages like funeral and Mass cards, newspaper clippings, recipes...
The best way to go about searching for a family Bible is to ask your relatives (someone may remember seeing one but not know what happened to it) and then trace various branches of your family for other living people to ask.
Added- sometimes the details recorded in a Bible have been transcribed (I found records dating back to 1600s in a Canadian archive).
