Does my visa expire along with my passport?

For the UK, you don't need to apply for a new visa if your passport expires. You can either travel with both your old passport (with the visa in it) and your new passport or pay a fee to transfer the visa to the new passport. But that's only true if the visa itself is still valid.

On the other hand, if the visa has an expiration date, then it's almost certainly not valid beyond that date (and it's possible the consulate chose the end of your passport's validity to set the visa's expiration date but it does not make a difference). If you still qualify for your original visa (say your family still lives in the UK), I don't think there is anything stopping you from applying for a new visa of the same type once you have a new passport but I guess you will need to go through the whole procedure again.

Note that visas allowing their holders to stay indefinitely are sometimes intended to be turned into something else after you entered the country. For example, you could apply to settle after entering on a “family of settled person visa” or apply for a UK residence card after staying on an EEA family permit.