Yes, lightning bolt affects everyone in its line of effect
Creatures do not block line-of-effect, so areas extend through them
A line area effect targets all squares (within the limits of its range and shape) to which the effect has line-of-effect.
Magic Overview > Aiming a Spell > Area > Cone, Cylinder, Line, or Sphere
A line-shaped spell shoots away from you in a line in the direction you designate. It starts from any corner of your square and extends to the limit of its range or until it strikes a barrier that blocks line of effect.
Note that, in the absence of such a blocker,
A line-shaped spell affects all creatures in squares through which the line passes.
So already there is a notion that multiple creatures can be included in the area, implying that they do not block the effect. But we can do better than an implication.
Glossary > Line of Effect
A line of effect is a straight, unblocked path that indicates what a spell can affect. A line of effect is canceled by a solid barrier. It's like line of sight for ranged weapons, except that it's not blocked by fog, darkness, and other factors that limit normal sight. A line of effect starts from any corner of your square and extends to the limit of its range or until it strikes a barrier that would block it. A line-shaped spell affects all creatures in squares through which the line passes.
Here we have the text explicitly stating that the line of effect passes through creatures. A creature is therefore not a solid barrier.
What Happens if the Rider undergoes forced movement?
PHB pg 198 also has a partially relevant sentence:
If you’re knocked prone while mounted, you must make the same saving throw.
While none of your above examples involve the 'Prone' status affliction, we can reason that under normal circumstances, if something were about to try and force move you (effectively off your mount), then you would be subject to the same saving throw to remain mounted as outlined above as you would if the mount were also being forcefully moved. (DC 10 Dex).
What about the military saddle?!
If you add in the military saddle, then you would have advantage on the DC 10 Dexterity saving throw to remain mounted.
This is an example of 'Specific beats general' rule of thumb. Were the general rule is found in spells like Thunderwave, that do not allow you save, they just work, and push you up to 15 feet; and the specific rule is what happens when you are mounted and you (or your mount) are forcefully moved while using a military saddle.
What about being grappled in your military saddle?
The grapple rules are on PHB pg 195:
"...you try to seize the target by making a grapple check, a Strength (Athletics) check contested by the target’s Strength (Athletics) or Dexterity (Acrobatics) check (the target chooses the ability to use). If you succeed, you subject the target to the grappled condition..."
Grappled: Your speed becomes 0...
Where as the other examples are doing something to impact you; a shove, a shockwave, a hard attack, the person grappling you is physically picking you up and moving you against your will (even though your speed is 0). So I would personally rule that you gain advantage on the check (because of the saddle) to avoid the grapple condition but once you fail and become grappled, you are at the mercy of your captor. You can continue to save and break free of the grapple, but you get no more advantage, and can do nothing to control being forcefully moved out of the saddle as you are grappled and no longer able to hold on to anything.
Application of the above example: Let's say a Kraken tried to grappled you (shame on you for riding your horse on the ship deck!), you could try and hold on to the saddle and fight away the tentacles, but if they overcame your resistance and wrapped themselves around you; you would be effectively helpless to do anything that would gain you advantage once it tried to pull you out of the saddle and see how tasty you are. You struggle as you are lifted out of your mount and over the water, drawing closer to the creatures maw, but you break free on your own just before you become Kraken food.
Best Answer
RAW is unclear about whether a mount and rider occupy the same space
One important question is whether a mount and its rider are considered to be occupying the same space, or whether the rider occupies a space above the mount's space. If they occupy the same space, then any area of effect that hits the mount can also be aimed in such a way that it hits the rider (although it is not guaranteed, since the mount is one size class larger, so it may occupy more space than its rider). If a rider is considered to be occupying a space above their mount, Then a horizontal area of effect, such as a Lightning Bolt fired approximately horizontally, will hit either one or the other but not both. That is, you could aim exactly at ground level and hit the mount, or aim at a slight upward angle and hit the rider on top of the mount.
Unfortunately, I can't find anything in the rules that specifies which of the above is the correct interpretation. The rules explain how to mount or dismount a creature, but they don't specify what that means in terms of relative positioning. This may be because there are a wide variety of possible shapes and sizes of mount (and rider!), so a general rule for all mounts might not make sense. Hence, the DM will need to make a ruling for a particular case.
Personally, I would rule that in the case of a griffon and humanoid rider, there is more than enough overlap between a rider and mount that there would be no problem hitting both with the same Lightning Bolt spell, regardless of the angle it is fired from.
Angle matters for line spells
Since you mentioned that the mount in question was flying, and presumably the wizard casting Lightning Bolt was on the ground, it is likely that the spell was fired at a potentially steep upward angle. In this case, even if the DM ruled that the rider occupied a space above the mount, the spell could still easily hit both of them if the angle is steep enough. In the most extreme case of firing 90 degrees straight up, the spell almost certainly will hit both of them.
The mount does not provide sufficient cover to protect the rider
The other possibility that might result in the rider not being hit by the spell is if the spell were entirely blocked by the mount, thus protecting the rider. However, a creature generally does not provide sufficient cover to block a spell. Generally, creatures provide at most half cover:
When shooting at a rider on top of a mount that is flying above you, it would be reasonable to grant the rider half cover (and to apply the optional rule about hitting the cover on a miss). However, as described in the section on areas of effect, only total cover is sufficient to block a spell's effect:
Hence, as long as the Lightning Bolt passes through both the mount's and the rider's spaces, it will hit both of them, and neither one will block the spell from hitting the other.