Why is Daniel Striped Tiger a father while Prince Tuesday's a teenager? Because tigers naturally age faster then humans.
It could also be because Daniel was of school age before Prince Tuesday was even born.
When Prince Tuesday was born, Daniel Tiger was already of school age. By the time Mister Rogers' Neighborhood ended, Prince Tuesday was himself of school age. Compulsory school age is usually six years old, so Prince Tuesday was six when Mister Rogers ended, and Daniel Tiger's age was presumably adjusted to reflect the initial age gap between them. The franchise's time skip lines up to reflect the eleven years between the final episode of Mr. Rogers and Daniel Tiger's debut. That makes Dad Tiger about twenty-three years old and Prince Tuesday about seventeen years old as of Daniel Tiger's Neighborhood.
Why does Prince Tuesday use the Affectionate Nickname "Little D" for Daniel? Well, Daniel is named after his father, who Prince Tuesday knew and called by his first name when they were both younger and thus the main character of this show is "Little Daniel" which neatly shortens to "Little D".
If Daniel Tiger's Neighborhood is a continuation of Mister Rogers Neighborhood, why don't the characters egg knowledge directly his life and death? Because the Land of Make Believe is an imaginary place where he never existed.
Sort of a Fridge Heartwarming/Awesome moment: In "Prince Wednesday's Happy Birthday", Daniel not only, as he pointed out, finally got cake that wasn't "smooshed", he finally got to have some of Prince Wednesday's strawberry birthday cake that he missed out on in "Daniel Gets a Cold".
In "Daniel Wants to Be Alone", Katerina thinks the cardboard box they're pretending is a castle looks "cosy", crawls in, and sits on the pillows in there. While it's meant to be because she wanted to be alone, maybe it's also because her cat side is showing.
In "Daniel's Bicycle", Chrissy is said to be very strong even though she needs crutches. Maybe she developed greater-than-average upper body strength to make up for her weak legs.
In the licensed game "Nature Walk", you get to choose between either of Daniel's parents to accompany him on said walk. Why can't they both go? Well, someone has to stay behind and watch Margaret.
Once an Episode Daniel Tiger, a character from the Land of Make-Believe, asks the viewer to make-believe with him. That's some Inception-level madness.
Why is the CROWN PRINCE of a Kingdom working as a waiter in a restaurant?!
Because he's not the king yet. He looks just old enough to be asking dad for the car keys.
If the question is why he has a mundane job rather than studying politics and diplomacy, perhaps King Friday thought his son would become a better ruler if he experienced what being a common worker is like? And of course the Doylist explanation is that it's a show for little children, who can identify with Wednesday if they have an older sibling with a job or a sibling who is closer to normal in general.
Tuesday may have wanted to see for himself what honest work is like.
The Land of Make Believe and all of the characters that inhabit it were once only in the imagination of Mr. Rogers. Many of the characters are children or extensions of characters that inhabited the original Land of Make Believe. As Fred Rogers died in 2003, and for there to be a timeline that extends past the run of Mr. Rogers' Neighborhood, we have to assume that Daniel Tiger's Neighborhood takes place within the imagination of a dead man.
For a more Heartwarming explanation- perhaps The Land of Make-Believe was brought back to life by the love of others who wanted to continue showing people that sweetness and magic.
Daniel has to take the school's pet rabbit Snowball home and take care of him. He's got to take Snowball home in a little cage. Daniel says he's never been to a rabbit's house before. Then there's an Imagine Spot of Snowball and a whole family of rabbits partying together, but then things get back to reality, and Snowball has only his cage. He's apart from his family.
Kerry: Remember when we were real little? ... Mom and Dad used to go out and we wouldn't want them to, and they'd say "Well, what do you know about mommies and daddies?" And we would say... "They always come back."
Bridget: Unless they collapse in Aisle 3 at the stupid grocery store.
