AHEM, for some backstory... Way back in the Third Astral Era, Robin's soul resided in Soph Ahm, an ancient Meracydian miqo'te who was found by Bahamut as an abandoned kit, and was raised as part of his and Tiamat's brood. Bahamut would eventually gift a large amount of his aether to Soph before falling in battle. Soph has his OWN story, which encompasses his journey to aid the people of Meracydia in summoning their eikons to arm themselves against the invading Allagan Empire, but for now what we need to know is that the original Soph was eventually captured by Allag and kickstarted the events of the Fourth Umbral Calamity. Before the Crystal Tower was swallowed by the earth, Desch (who Soph grew to trust) escaped with an Allagan clone pod containing a genetic copy of Soph made by Amon, infused with dragon blood and all, which ended up safely in Corvos. This clone would awaken some centuries later and continue Soph's dragon-touched bloodline.

skip ahead 5,000 or so years and Soph's soul is reborn as Robin, who ALSO happens to be a distant blood descendant of the aforementioned clone. This grants him easy access to Bahamut's aether and all the dreadwyrmy goodness (and destruction) that comes with it :) Tiamat senses something familiar about Robin, as well as this strange connection to his aether; upon having a revelation in Paglth'an, she comes to see him as her child, just as Soph was.

tl;dr: Robin sees Tiamat as a motherly figure and the key to explaining some of his inherited burdens, but his relationship with Bahamut is a bit more complicated. The Dreadwyrm's power has been one of those aforementioned burdens on him since he was a wee lad, and has caused much damage to himself and those around him. But they're chill now