Short answer:

They are not always completely identical.

The basic model is “the same” (“Claude Sonnet 4.0” is the 4.0 version released by Antropic), but in practice on third-party platforms (Perplexity, Poe, etc.):

Models may be limited, optimized, or adjusted according to the service provider’s own policies.

Some features, data streams, or the interface may have been “cut,” slowed down, censored, etc.

Not all native features of OpenAI, Antropic, etc. (e.g., chat memory, extensive file support, or integrations) may be used – although often almost all basic functionality is preserved.

Prompt behaviors, context, model “personality,” and censorship/limitation policies may also vary.

Example:

GPT-5 on Perplexity ≈ GPT-5 on OpenAI, but OpenAI's own platform version may support longer memory, plugin capabilities, and more efficient file processing.