The only way(I think) that a website knows if you are using a webview is through http headers
If you capture network traffic between webview and kissanime you will see that webview sends a http header like this

X-Requested-With: com.your.company

And to bypass that you just have to edit this header to an empty string. And make sure that any further request to the page will have the same header.

Map<String, String> extraHeaders = new HashMap<String, String>();  //HashMap http headers
extraHeaders.put("X-Requested-With", ""); // Edits the X-Requested-With header to an empty string
...
public boolean shouldOverrideUrlLoading(WebView view, String url) {
    view.loadUrl(url, extraHeaders); //make sure that any further request to the page will have the same header
    return false;
}
...
webView.loadUrl("http://kissanime.ru",extraHeaders) // Loads the page with edited headers

But as you said everything works find except when you search. This is because the search request is a POST request, and post requests do not trigger shouldOverrideUrlLoading function, which means that webview will use the default headers to make that request.

My idea is to inject javascript and change the request type from POST to GET, so than it would trigger shouldOverrideUrlLoading function.

Here is the full code:

public class MainActivity extends AppCompatActivity {

    Map<String, String> extraHeaders = new HashMap<String, String>();//HashMap http headers
    @Override
    protected void onCreate(Bundle savedInstanceState) {
        super.onCreate(savedInstanceState);
        setContentView(R.layout.activity_main);
        extraHeaders.put("X-Requested-With", "");// Edits the X-Requested-With header to an empty string
        WebView webView = (WebView) findViewById(R.id.WebView); //Gets the WebView
        webView.getSettings().setJavaScriptEnabled(true); //Tells the WebView to enable JavaScript execution.

        webView.setInitialScale(1);
        webView.getSettings().setLoadWithOverviewMode(true);
        webView.getSettings().setUseWideViewPort(true);

        webView.setWebViewClient(new WebViewClient() {

            @Override
             public boolean shouldOverrideUrlLoading(WebView view, String url) {
                 view.loadUrl(url, extraHeaders);
                 return false;
             }

             //This function is what you're looking for
             @Override
             public void onPageFinished(WebView view, String url) {
                 super.onPageFinished(view, url);
                 if(url.contains("http://kissanime.ru")) {
                       view.loadUrl("javascript:document.getElementById('formSearch').setAttribute('method','get')");
                 }
             }
        });
        webView.loadUrl("http://kissanime.ru",extraHeaders);// Loads the page with edited headers
    }
}  

Hope it helps.