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.
Chrissie's horse is named Peaches. Daniel is allergic to peaches. Therefore, Chrissie's horse Peaches is to be feared, and Daniel was especially justified in asking his father to hold his hand while he rode her.