Corrected entry: Diane says that her parents named her Diane because they liked the show "Cheers." However, she recently turned 35, and this episode was created in 2015. She was born in 1980, but the first "Cheers" episode was in 1982.
Corrected entry: When Diane and Mr. Peanutbutter accidentally met at the bar, they first make eye contact. But Mr. PeanutButter being a dog, he would have recognized her smell as soon as he entered the room. (00:16:40)
Correction: The animal characters in this universe do not show all the traits they would in the real world. Fish for example are shown walking around on dry land. For the most part the animal characters only share traits with their real world inspiration when it's funny, such as Mr. Peanutbutter sticking his ears up when he's excited.
Mr. Peanutbutter's Boos - S5-E8
Corrected entry: Diane talks to Bojack during the 2009 Halloween party. He is using the same red smartphone that he has in the present day. This is not impossible, but it seems unlikely for a celebrity to keep using the same phone for nine years, considering how many non-celebrities don't even keep the same phone that long.
Correction: You said yourself it is not impossible. He could also just have a similar looking phone. Many iPhones look pretty similar to previous models.
Correction: In season 6, Diane says that her parents didn't name her until she was four years old. Likely a line written to fix this mistake, but it's still canonical and explains the apparent discontinuity.