Sounds like we need an extended JSON with the express intent of conveying common extended values and rich objects: DateTime instants (with calendar system & timezone), Decimal, BigInt, etc.
If you do want the interchange format to be the one deserializing into specific runtime data structures, use YAML. YAML's tag syntax allows you to run arbitrary code inside YAML, which can be used for what you want.
This would probably best exist as a well-known wrapper around JSON itself.
Few people seem to use it outside of Amazon tho
CBOR (Concise Binary Object Representation) has JSON-like semantics with type extension support; with built in type extensions its much easier to get some agreement about registering certain magic type IDs to mean certain things. for example from a random google search for "cbor datetime" https://j-richter.github.io/CBOR/date.html; there's an IANA registry of type IDs: https://www.iana.org/assignments/cbor-tags/cbor-tags.xhtml
however, it is binary.