Support react 19

[Mister-Hope]

Describe the problem

React 19 has been published for a while, it would be great if docsearch and support react19, so users can get rid of issues caused by multiple versions of react.

[slorber]

Docusaurus website is using DocSearch with React 19 and it seems to work fine: https://docusaurus.io/

Maybe just allowing ^19 on peerDeps is enough?

[notaphplover]

This is becoming a bottleneck for docusaurus users using the default theme. Since docusaurus 3.7, React 19 is supported, but depending on @docsearch/react as an outcome of depending on @docusaurus/theme-search-algolia generates a peer dependency conflict:

@docusaurus/theme-search-algolia 3.7.0
    └─┬ @docsearch/react 3.8.2
      ├── ✕ unmet peer react@">= 16.8.0 < 19.0.0": found 19.0.0
      ├── ✕ unmet peer @types/react@">= 16.8.0 < 19.0.0": found 19.0.2
      └── ✕ unmet peer react-dom@">= 16.8.0 < 19.0.0": found 19.0.0

[tats-u]

#2388

Some fixes seem to be required for React 19.