I think the below quote pretty much sums up that any answer to the question is going to be mostly speculation:
We shall not pretend to any understanding of the bond between dragon and dragonrider; wiser heads have pondered that mystery for centuries. We do know, however, that dragons are not horses, to be ridden by any man who throws a saddle on their back.
Fire & Blood
So really we don't know what else can break a bond between the rider and the dragon. However, for the most part a dragon might not know the difference between their rider dying and their rider abandoning them. Note that Seasmoke was Laenor's pride and passion; it is strange of him to abandon the dragon and so Seasmoke may just assume Laenor is dead.
To further this point note that in the books Laenor is actually dead-dead. He is killed with an audience in the middle of a fair. The addition of his surviving in the show seems to be to add a somewhat happy end for him rather than to make any sense. So we can probably ignore any canon inconsistencies that follow from this.
The Lord of the Tides and his lady were still in mourning for their beloved daughter when the Stranger came again, to carry off their son. Ser Laenor Velaryon, husband to the Princess Rhaenyra and the putative father of her children, was slain whilst attending a fair in Spicetown, stabbed to death by his friend and companion Ser Qarl Correy. The two men had been quarreling loudly before blades were drawn, merchants at the fair told Lord Velaryon when he came to collect his son’s body. Correy had fled by then, wounding several men who tried to hinder him. Some claimed a ship had been waiting for him offshore. He was never seen again.
Fire & Blood
However, one point contrary to this is the case of Helaena Targaryen and Dreamfyre. Dreamfyre is said to have sensed Helaena's death and acted out in pain. So, it is possible for the two to share a close bond such that they are aware of each other. Though this isn't necessarily the case with all dragons.
At the moment of her death, across the city atop the Hill of Rhaenys, her dragon, Dreamfyre, rose suddenly with a roar that shook the Dragonpit, snapping two of the chains that bound her.
Fire & Blood
Lastly, it is worth noting that Seasmoke remains riderless for roughly a decade in the books. We are yet to see how the timeline plays out in the show of course. If that is the case then Seasmoke really would feel abandoned and possibly believe Laenor to be dead. And even then Seasmoke won't let just anyone attempt to mount him; with several attempts at people trying to claim him and failing with some of those perishing.
Dragons are not horses. They do not easily accept men upon their backs, and when angered or threatened, they attack. Munkun’s True Telling tells us that sixteen men lost their lives during the Sowing. Three times that number were burned or maimed. Steffon Darklyn was burned to death whilst attempting to mount the dragon Seasmoke. Lord Gormon Massey suffered the same fate when approaching Vermithor.
Fire & Blood
Just to tack on the end because you've added it to the question now but there have been several instances of people trying ride a dragon that still has a rider; they usually do not end up well. In the case of Joffrey Velaryon trying to ride Syrax, Rhaenyra Targaryen's dragon, he ended up being thrown to the floor and dying.
Syrax was the queen’s dragon. She had never known another rider. Though Prince Joffrey was known to her by sight and scent, a familiar presence whose fumbling at her chains excited no alarm, the great yellow she-dragon wanted no part of him astride her. In his haste to be away before he could be stopped, the prince had vaulted onto Syrax without benefit of saddle or whip. His intent, we must presume, was either to fly Syrax into battle or, more likely, to cross the city to the Dragonpit and his own Tyraxes. Mayhaps he meant to loose the other pit dragons as well.
Joffrey never reached the Hill of Rhaenys. Once in the air, Syrax twisted beneath him, fighting to be free of this unfamiliar rider. And from below, stones and spears and arrows flew at him from the hands of the Shepherd’s blood-soaked lambs, maddening the dragon even further. Two hundred feet above Flea Bottom, Prince Joffrey slid from the dragon’s back and plunged to the earth.
Fire & Blood
