Is my date of birth stored anywhere?

No. The calculation runs on the page from what you type, nothing is saved to an account and there is no cookie. Clear the form and nothing remains.

Does this mean I am definitely eligible?

No — it means the vacancy is worth opening. Only the official notification decides eligibility, and every result here links to it.

Is age relaxation the same for every exam?

No. The relaxation shown is the common central-government pattern (OBC 3 years, SC/ST 5 years, PwBD 10 years). State police, defence and teaching recruitments differ, and the notification's own cut-off date, not today, is what counts.

Which government jobs am I eligible for?

Enter your date of birth, qualification and state to see every currently-open sarkari vacancy you can apply for, sorted by closing date, with the age limit worked out for your category.