If there is no absolute time, how can we say the Big Bang was 13.8 billion years ago?
First, you're quite wrong about Bob --- when he returns, he shares Alice's frame, and so agrees with Alice that the universe is 14.8 billion years old --- and that he once spent one billion of those years traveling, during which time he aged only .1 billion years, though this has nothing to do with how long ago the Big Bang occurred.
The person you want to ask about is not Bob, but Carla, who has been zipping along at .99c (relative to the earth) for several billion years, and is currently just passing the earth. She and Alice will certainly not share a time coordinate.
So it still makes sense to ask according to whom is the universe 14.8 billion years old? Roughly the answer is: According to an observer who has been traveling with along with a galaxy. In reasonably simple models of relativity, it doesn't matter which galaxy you pick. The motion of the galaxies (plural) does give a global time coordinate (though there are still perfectly legitimate frames, like Carla's, that do not incorporate that time coordinate).
Nobody believes those models are exactly accurate, but they can still be close enough to the truth that one can speak of the age of the universe as calculated in one of those models, and attribute some meaning to it.
You are correct, there is no absolute time in the universe. This is as per SR, time is relative.
Now to whom is the universe 13.8 billion years old? To a comoving observer with zero comoving velocity (peculiar velocity).
The "age of the Universe" of about 14Gyr you frequently hear about is a good approximation for any observer whose peculiar velocity is non-relativistic at all times. In practice these are the only observers we're interested in, since peculiar velocities for any bulk object (like galaxies) tend to be non-relativistic. If you happened to be interested in the time experienced by a relativistic particle since the beginning of the Universe, it wouldn't be terribly hard to calculate.
The age of the universe
Now this is only true for nonrelativistic observers. So you are correct, there is no universal time, but we usually use the CMB (Cosmic Microwave Background) as a background reference frame.
Understanding the CMB background as a reference frame
Relativity has ways to make time pass arbitrarily slow, but no ways to make it pass arbitrarily fast. So if you are uncomfortable with defining what it means to "move with the universe", there is another way. You can define the age of the universe as the longest possible time any observer can need to get from the big bang to our current position in space-time. (Without cheating by using wormholes and other time-traveling shenanigans) The final result would of course effectively be the same.