We can use a trigger on the AgentWork entity to update the Case Owner to Chat agent when the Chat is accepted in Salesforce.

Sample Trigger:

trigger AgentWork on AgentWork ( after update ) {
    
    Set < Id > setChatTranscriptIds = new Set < Id >();
    Map < Id, Id > mapChatIdAgentId = new Map < Id, Id >();
    
    for ( AgentWork objAW : trigger.new ) {
    
        /*
            Checking whether the Work Item routed is Chat and the status is Opened.
            Status Opened means the agent has accepted the Chat.
        */
        if ( objAW.Status == 'Opened' && String.valueOf( objAW.WorkItemId ).startsWith( '570' ) ) {
        
            setChatTranscriptIds.add( objAW.WorkItemId );
            mapChatIdAgentId.put( objAW.WorkItemId, objAW.UserId );
        
        }
    
    }
    
    if ( setChatTranscriptIds.size() > 0 ) {
    
        List < Case > listCases = new List < Case >();
    
        for ( LiveChatTranscript objLCT : [ SELECT Id, CaseId FROM LiveChatTranscript WHERE Id IN: setChatTranscriptIds ] ) {
            
            if ( String.isNotBlank( objLCT.CaseId ) ) {
            
                /*
                    Updating Case Owner to Chat Agent
                */
                listCases.add( new Case(
                    Id = objLCT.CaseId,
                    OwnerId = mapChatIdAgentId.get( objLCT.CaseId )
                ) );
            
            }
        
        }
        
        if ( listCases.size() > 0 ) {
        
            update listCases;
            
        }
    
    }

}

Output: