The Morningstar symbol from the game has different colours/design (it's a hollow 5 pointed star, where the one in American Dad if filled with a darker colour to the outline), and correct me if I'm wrong, but I don't think
Saints Row: The Third was the first time something with star in it's name also used a star for it's symbol, so it's just a coincidental use of the word star and the star symbol.
The five pointed star and variation of it is quite an old and popular symbol and has been used by many organisations for a long time, even before the first
Saints Row game... hell even before the PS1.
It is also quite popular with Communism, as you can see above (with the symbol from
Saints Row in the middle from comparison, so I can say with almost certainty the symbol in American Dad was just a coincidental use of the star symbol to fit with the name Dark Star (as the inner part of the star symbol is dark).
I also think the symbol in
Saints Row: The Third is actually misused and doesn't fit the name, as most traditional symbols for the Morningstar aren't 5 pointed stars and are shaped and oriented differently, so it's more likely that the name Morningstar was an afterthought when they'd already came up with the Communist/Russian imaginary and symbolism for the gang and it's Red Star inspired symbol (which isn't actually a Morningstar).
If this is true it means that the Darkstar reference in American Dad certainly isn't a reference to
Saints Row: The Third because American Dad actually uses a symbol that is symbolic of it's name, whereas
Saints Row uses a Red Star to represent Morningstar, which would be a bit like using an octagon to represent an organisation called The Circle.