The conversation implies Rick is not living with the same daughter Beth he begat, and that Rick's Beth is dead.
Whether Rick is living with a cloned Beth (prior to the deliberate Beth cloning in Season 4), or with a Beth from a different dimension, is yet to be revealed, but the clues exist and are explained below. Also, I refer to it as "dimension", not "timeline" because that's how the show refers to it.
The Rick we follow for most episodes is purportedly Rick C-137. Morty believes C-137 is the name of his home dimension, which was Cronenberged in "Rick Potion #9" (Season 1, Episode 6). It is not; we'll label Morty's origin dimension as C-BERG and Rick's origin dimension as C-137.
In "Rick Potion #9", Rick and Morty swap from dimension C-BERG, which becomes unliveable, into a parallel dimension we'll label C-NEW. In C-NEW, history was identical to C-BERG except (1) the events of S1E6 did not happen, and (2) the Rick and Morty living in C-NEW accidentally kill themselves.
In "The Rickshank Rickdemption" (Season 3, Episode 1), while under interrogation, Rick reveals a memory of the death of his wife Diane and child Beth, an event which pushes him to finish creating his dimension portal gun. Rick later implies this memory was fabricated to trap his interrogator, but this may not be true and could reveal how Rick C-137 may have lost Beth C-137.
Also in "The Rickshank Rickdemption", Morty C-BERG revisits dimension C-BERG (with Summer C-NEW in tow) and encounters parallel Ricks from the Council of Ricks. When he identifies himself as "Morty C-137", the Ricks give him a split-second, blink and you'll miss it, double-look. This implies early on that C-BERG is not C-137; there is no Morty from C-137 because Beth C-137 never survived to adulthood; Rick C-137's memory is true.
In "Rickternal Sunshine of the Spotless Mort" (Season 5, Episode 8), Rick C-137 enters Bird Person C-NEW's memory and encounters a younger Rick created from Bird Person's memories of the Rick who originally lived in C-NEW; Memory Rick.
Memory Rick not only says, "You live with a version of our dead daughter" he follows with "You're one of those creeps who moves in with abandoned adult Beth." This implies Bird Person believes the Rick he knew from C-NEW (1) lost his original Beth and (2) is not originally from C-NEW and (3) therefore is not the original father to Beth C-NEW. Bird Person knew a Rick we'll label as C-NOTNEW. Notably, throughout the show's episodes, he has been interacting with Rick C-137.
Since dimension C-NEW was "identical enough" in history to C-BERG, and Memory Rick pluralizes "creeps", one can surmise Beth dying and Rick swapping to a new family happened with not only Rick C-NOTNEW and Rick C-137, but many other Ricks.
In summary: Beth C-137 died as a child, and Rick C-137 eventually moved to C-BERG to live with the adult Beth C-BERG and recruited Morty C-BERG. Rick C-BERG abandoned Beth C-BERG, leaving a hole for Rick C-137 to fill. What happened to Rick C-BERG, we do not know other than Memory Rick implies Beth was "abandoned"; perhaps the Ricks that abandon Beths join the Council of Ricks.
Ultimately, Rick C-137 left C-137 for C-BERG, joined the C-BERG family, then Rick C-137 and Morty C-BERG moved to C-NEW, replacing Rick C-NOTNEW and Morty C-NEW to live with C-NEW Beth, Summer, and Jerry.
Notably, in "Mortynight Run" (Season 2, Episode 2), Jerry C-NEW was swapped with another dimension's Jerry when departing the Jerryboree daycare; therefore Rick C-137 and Morty C-BERG live with Beth C-NEW (who may be a Beth clone after Season 4), Summer C-NEW, and Jerry C-???.
The implication is that Bird Person C-NEW believes Rick C-NEW is actually Rick from dimension C-NOTNEW where Beth C-NOTNEW was killed, and is living with the daughter of a Rick C-NEW who abandoned Beth C-NEW. Bird Person does not know Rick C-137 replaced Rick C-NOTNEW, however the histories of Rick C-NOTNEW and Rick C-137 are similar enough that it probably does not matter. Further implication is that Morty C-BERG does not realize there never was a Morty in C-137, and that his Rick did not originate from C-BERG.
To take it one step further, in "Rixty Minutes" (Season 1, Episode 8), Summer C-NEW fails to consistently find a parallel self using the Inter-Dimensional Goggles; while this is chalked-up in-episode as her parents getting an abortion in those realities, it's entirely possible Summer didn't exist in some of them because the Beth of those dimensions was killed in childhood.