Is it permissible to pray two raka’ah (units) prayer on our birthdays to remember the years we have spent on earth, and use the medium to thank Allah and seek forgiveness from Him? Is it a crime to mark the birthday as such?

If this is something that was done by the Prophet (ﷺ), we should follow it. However, the Prophet (ﷺ) showed his gratefulness to Allah by fasting, not praying two raka’ah units prayer. It must be known that for any act of worship to be accepted by Allah, it must fulfill two conditions. First is it must be done sincerely for the sake of Allah. Second is that it must follow the Sunnah of the Prophet (ﷺ).

Allah knows best.