(re)Captcha is just a tool you are using to identify if the user is human and users don't care how you try to do that as long as it isn't bugging them too much. The tool itself doesn't mention it is called reCaptcha to the user (only as a disclaimer), it just says "Select all squares with traffic lights". Most people don't know why it is needed or only understand it partially, but this way it is acceptably accessible.
With the error messages it is advisable to follow that principle and keep that acceptably accessible too. It isn't relevant to call it reCaptcha and it isn't relevant to mention if it did fail, expire, explode or whatever went wrong in the backend. All what is relevant is what to do next: try again or contact support. reCaptcha is helping you, not your users. When it fails you are dealing with a robot or you're punishing a user unnecessarily. In that case you want to be informed about the error and give (human) users a link and/or phone number to contact support. Also keep the error message short and functional and leave the appologies to support.