The Flock spell restricts the creatures summoned
The Flock of Familiars spell does say:
Each familiar uses the same rules and options for a familiar conjured by the find familiar spell.
However, immediately preceding that, it also says:
You temporarily summon three familiars – spirits that take animal forms of your choice.
Spell descriptions are rules, and spells do what they say they do. Flock of Familiars restricts the familiars summoned by it to wearing the forms of animals, even though they are spirits. This restriction exists independently of the reference to find familiar, and it still applies when you use the Flock spell even if some other feature changes what happens when you cast the find familiar spell.
All of the animal forms listed in the find familiar spell itself have the Type stat of 'Beasts', which the Monster Manual (p. 6) describes as:
nonhumanoid creatures that are a natural part of the fantasy ecology...Beasts include all varieties of ordinary animals, dinosaurs, and giant versions of animals.
Of the four forms accessible to Warlocks with Pact of the Chain, none of them are Beasts:
Imps and Quasits are Fiends, Pseudodragons are Dragons, and Sprites are Fey.
Thus the Flock of Familiars spell itself tells us that it cannot be used to summon a familiar in the additional forms that are permitted to Pact of the Chain Warlocks.
Pact of the Chain overrides Find Familiar, but not Flock
Pack of the Chain says (emphasis mine):
You learn the Find Familiar spell and can cast it as a ritual. The spell doesn’t count against your number of Spells known.
When you cast the spell, you can choose one of the normal forms for your familiar or one of the following Special forms: Imp, Pseudodragon, Quasit, or Sprite.
The features permitted by the Pact of the Chain take effect when you cast the spell - but it is clear that "the spell" is find familiar, not flock of familiars. That is, Pact overrides the features of find familiar upon casting - the limitation of the forms to the list provided. But it does not override the limitations of flock of familiars - that the spirits be in the forms of animals - when you cast that spell.
Further, Flock of Familiars says:
Each familiar uses the same rules and options for a familiar conjured by the Find Familiar spell.
Flock says it uses the same rules for a familiar conjured by the find familiar spell (that is, the general one on-book), not that it uses the same rules (and rules changes) that apply "when you cast" the find familiar spell. Flock does not say that 'each familiar uses the same rules and options for a familiar conjured by you'.
Pact of the Chain allows you to change how the find familiar spell works for you, but it does not permit you to change the spell itself, which is what flock is referencing. Nor does your Pact permit you to change how the flock of familiars spell works for you. The limitation of the flock spell to summoning familiars that have animal forms is not overridden by your Pact.