Question 1. Is it correct to say in formalised ZFC that there is just one empty set?
ZFC requires a unique empty set, i.e. for ZFC there is just one empty set. This is due to the Extensionality axiom of ZFC, which identifies two sets whenever they have the same elements - formally:
∀ x ∀ y [∀ z [z ∈ x ↔ z ∈ y] → x = y]
and the Null Set axiom, which requires the existence of an empty set (i.e. a set containing no other set) - formally:
∃ x ∀ y [y ∉ x]
Wikipedia.Axiom of extensionality
nLab.Axiom of extensionality
Question 2. Is it correct to say that in the categorical set theory, say ETCS, that there are many empty sets but they are all isomorphic?
ETCS requires at least one and allows several initial objects alias empty sets. This is because an initial object 0 (that can be thought of as representing an empty set) is defined by a universal property, namely that for any object x there is exactly one mormhism from 0 to x - formally:
∀ x ∃! 0 → x
(the arrow here is a morphism, not a deduction) and is thus defined up to (unique) isomorphism. However since the initial objects are isomorphic one speaks of the initial object.
Two initial objects 0 and 0' must be isomorphic: since 0 is initial it exists 0 → 0', and since 0' is initial it exists 0' → 0. Then the composition 0 → 0' → 0 must be the identity on 0 (since 0 is initial, there can by definition just be one mormhism 0 → 0 but this has to be the identity, which is given in any category for any object). Analogue argumentation shows that 0' → 0 → 0' must be the identity on 0'. So 0 and 0' are isomorphic (what we just showed is the definition of two objects being isomorphic: a and b are isomorphic if there are two morphisms f: a → b and g: b → a and gf = id: a → a and fg = id: b → b.
Wikipedia.Initial and terminal objects
Example (Sets of cars).
If you would regard sets of cars, then any car, that is not containing other cars (so excluded loaded car transporters or cars with matchbox cars inside), is an empty set. ZFC regards all those cars as identical - thus there is only one car that is not containing other cars. ETCS allows to distinguish them, but regards them as isomorphic.
Question *. Which is the more accurate view?
My personal opinion. Asking for accuracy is perhaps not the right approach here. One could instead ask which concept to work with. This depends on your criteria:
(1) In case you are interested in the study of well-orders and the cumulative hierarchy, you might want to work with ZFC.
(2) In case you are interested in mathematics apart from that and how sets are used there, you should probably have a look at category theory and ETCS rather than ZFC.
Leinster.Rethinking set theory