Following Saki's signature style, even the singularly ridiculous name of the Pigeoncotes hints at their similarly peculiar personalities.
A parody of the shallow English upper classes, the Pigeoncotes are a male-female married couple celebrating their 25th wedding anniversary. This lets us infer that they are middle-aged, or entering their middle-adulthood. Based on the fact that the husband, Peter Pigeoncote has a relative who inherited the family baronetcy, we can also infer that the family has pedigree, name, and historical relevance.
This family member who inherited the baronetcy, Wilfrid, has also inherited a lot of money. Therefore, this means that the Pigeoncotes are also rich. Evidence of this is given in the number of gifts the married couple receives on their anniversary. It is also evident in the fact that the couple has an estate big enough for family to feel comfortable staying at while passing through from the country to the city.
One indirect characterization of the Pigeoncotes is that they are judgmental. Their fixation with one of their relatives, a kleptomaniac cousin named Wilfrid, led them to confuse him with the Wilfrid who inherited the baronetcy and who also came to visit them, even bringing with him a gift for their anniversary. This early judgment led them to assume that the visiting Wilfrid had stolen from them, only to find out that they were wrong.
A last indication of the shallow nature of the Pigeoncotes is that they are unwilling to admit to a mistake. Rather than "explain everything", like Mrs. Peter Pigeoncote tells her husband that she did, what she actually did do is tell Wilfrid, in secret, that Peter is a kleptomaniac, and that he (Peter) took Wilfrid's cream jug while the latter was asleep. This would have excused the fact that the Pigeoncotes went into Wilfrid's room to see if Wilfrid had taken one of the silver jugs that Peter claimed to be missing.
The jug was not missing; they just miscalculated how many they had. When they saw the one Wilfrid had purchased for them inside his bag, they assumed it was their missing jug and that Wilfrid had stolen it. They removed the jug from Wilfrid's bag and took it out at breakfast. When Wilfrid saw it, he assumed that the help of the house (servants) were thieves and took it from him. To save face, Mrs. Pigeoncote came up with the lie that Peter was a kleptomaniac, meaning that their judgmental nature came back to bite them.
Therefore, it is safe to describe the Pigeoncotes as shallow, judgmental, very rich, and very concerned about tainting their reputation as a "family name."