The system marks the response in the first survey completed when it gets chained to the next survey based on the survey chaining setup. In this case, you can set up Survey Chaining on the very last question of the first survey. This way, respondents can answer all the questions in the first survey, and their Response Status will be marked as Completed.

If you prefer not to set up Survey Chaining on the last question of the survey but in the mid-survey, you can consider using Auto Redirect as an alternative option for fulfilling this requirement. You can simply use the survey link of the second survey in Auto Redirect. Help file: https://www.questionpro.com/help/auto-redirect.html

In case you want to store the response ID of the first survey with the response of the second survey then you can pass the response ID in the redirect link through custom variable.

Screenshot: https://prnt.sc/VflN2aQ7I_DL