no

To the first question, or both?

And is this intuition or a clearly stated rule somewhere?

no to the first question. familiar isn't a class. depending on how you look at it, its not even a creature. its a class benefit, and the abilities the familiar has are benefits of having atained a certain level in a specific class. they get clearly defined bonuses and abilities listed on the familiar abilities table in the phb. if they got feats that would be listed on the table.

as to the second question, if the prestige class grants a level in spellcasting ability, i would have, in the past, allowed a familiar the next level's worth of abilities. but the rules seem pretty clear.

All familiars have special abilities (or impart abilities to their masters) depending on the master’s combined level in classes that grant familiars, as shown on the table below. The abilities given on the table are cumulative.

now that i've read that, i'd say no. i wouldn't.

to clarify, if the prestige class in question allows the summon familiar ability, or specifically states it grants another level of familiar ability, then i'd grant the level of ability. if it doesn't, i wouldn't.

Under normal situations it would, but Familiars are extensions of the spellcaster. If, for example, a Familiar has 3 HD before becoming a Familiar, it would have 2 feats, but otherwise ... no. Typically, also, what feats a Familiar gets are outlined in its description in the Monster Manual.

And Familiar advancement doesn't carry over to prestige classes unless they have the Obtain Familiar feat.