The COVID-19 pandemic has been shown to have impaired the mental health and well-being of young people. This study, for the first time, explores these aspects in young people with and without a migratory background during the extended course of the pandemic and restrictive measures, comparing two countries with a high COVID-19 prevalence: Austria and Turkey. Methods: The authors used the “Psychological General Well-being” index as part of an anonymous online survey with 3665 participants (ages 15–25), recruited from both countries during the first and the second waves of the pandemic, collecting data on individual experiences and problems encountered during the pandemic. Results: Mental health (b = 0.06, p < 0.023) and general psychological well-being worsened with the progression of the pandemic. Participants with financial problems had the most severe negative effect on mental health (b = 0.12, p < 0.001). Furthermore, females living in Turkey, both natives (b = −0.21, p < 0.001) and migrants (b = 0.25, p < 0.001), reported a more deteriorated mental health status over time. Conclusions: The extended pandemic duration and resultant “lockdown” restrictions have negatively affected the mental health of young people to varying degrees, depending on country of residence and migration background. A strong “recovery plan” that considers group-specific needs and vulnerabilities is urgently needed.