No the books and the TV adaptation are heading in completely different directions because the Show is well beyond the last Book A Dance With Dragons and the next book won't be releasing anytime in the foreseeable future.
A.T = J.S.?
Jon Snow in no way equates Aegon Targaryen in TV Show's universe. TV adaptation does not include Aegon's story and it doesn't seem like they would be doing it at all because in Season 7, it will be plausibly too late for that arc to be included in the broader picture. In any case Jon Snow does not equate Aegon because:
1. Aegon has support of Golden Company. Jon doesn't.
2. Aegon's first military campaign was aimed at House Baratheon of King's Landing. Jon on the other hand is focused on the North and is
currently crowned King in the North.
3. Aegon is son of Elia and Rhaegar while Jon is son of Lyanna and Rhaegar.
4. Varys is devoted to Aegon in the books while in the show he is completely devoted to Daenerys and is currently accompanying her to
5. Aegon's first major military conquest was Storm's End while Jon's is Winterfell.
6. Aegon might in fact not be Rhaegar's son after all. See my previous answer in this regard here.
That does not mean show has not done anything like that. Edric Storm's role was given to Gendry in TV adaptation but there is substantial evidence in support of this claim as many things which happened with Edric were presented to have been happened to Gendry such as:
- His blood was used for the leeches
- Melisandre pestered Stannis to kill the boy
- Finally the escape scene on a naval vessel with help of Ser Davos Seaworth.
As you can see, while one can easily see how Gendry = Edric in Show adaptation, there is nothing which suggests similar relationship between Jon and Aegon.
Will he propose a marital Alliance to Daenerys?
That would be speculation as this is about future works. It isn't unlikely and Daenerys would be predisposed towards the idea as she needs loyal allies on mainland Westeros.
In any case, Since Daenerys has three dragons, she might not be that much bothered about a marital alliance as she's got all she is going to need. She is equipped with everything her ancestor Aegon the Conqueror was when he conquered the seven Kingdoms with a small army, a fleet and three dragons. She is however lacking two dragonriders.
If it is revealed and somehow proven that Jon is in fact her nephew, she might be more than willing for the marital alliance provided they share the rule or Jon adopts a junior role in administration (Something like how Rhaenyra's consort Daemon adopted only the title of Protector of the Realm and was never referred to as King). Jon would then be the second head of the dragon.
Will that solve legal and lineage issues?
No, it would not.
But the thing is, as mentioned on another question which I can't remember, there is no Supreme Court of Westeros to enforce the law. Might is right.
The problems would still exist because:
Even if Jon is proven to be a son of Rhaegar and Lyanna, he will still
be a bastard as there is nothing to prove that Jon was born inside
wedlock. He will need to be legitimized by a royal decree to assume
the Targaryen or Stark name. Daenerys can do that as Queen of
But first, this has to be revealed to people of Westeros that Jon is
Son of Lyanna and Rhaegar. Only two people know this who are Bran and
Lord Howland Reed.
Will that unify everything?
No, not really.
Those who are against Targaryens or Starks will oppose their alliance anyways but they can be forced into submission easily because, you know, dragons.
In a feudal society, it is certain that there will always be some aristocrats who will be against the monarch. A perfect monarch who is loved by everyone has never existed.
Targaryen dynasty was not unanimously loved or supported even before Aerys II's coronation. There were many revolts and uprisings against the Targaryens which include:
- The War of Faith Militant
- First Dornish war of Independence1
- Second Dornish War of Independence2
- Stormlander revolt of Lord Lyonel Baratheon
- First Vulture King's revolt
- Second Vulture King's revolt
- Ironborn uprising of King Lodos
- Defiance of Ironborn under Dalton Greyjoy
- Defiance of Ironborn under Dagon Greyjoy
- Ironborn Uprising of Harren the Red
- Rebellions of Daeron the Daring imposters
- Blackfyre rebellions
- Peake Uprising3
- Defiance of Duskendale
- Robert's Rebellion4
Unification is something which is perfect and perfection does not exist in a universe whose arch-stone is characters being grey.
A common enemy can however provide a temporary unification. As it happens, there is a common enemy in shape of the White Walkers. Northmen appear to have utmost trust in leadership of Jon Snow in wake of his victory against the Boltons. Daenerys' own men have the same feelings about her, probably even stronger. But for other Lords and peasants of Westeros, both Daenerys and Jon will have to prove that they are the leaders behind whom the whole Westeros can stand together.
Update for Season 7
In Season 7 Finale, it was revealed Jon's real name is Aegon
Targaryen. Also previously it was revealed that Rhaegar got his marriage to Elia annulled and got married to someone else who can be none other than Lyanna. That makes him legitimate.
1. Queen Rhaenys was killed in action along with her dragon Meraxes
2. King Daeron I was killed in action.
3. King Maekar I was killed in action.
4. King Aerys II was killed along with his heir Prince Rhaegar and Rhaegar's children.