Thank you @LeonG for the reply.
I agree with what you wrote and my understanding is that if I wanted the payout instead of the holidays, I should have discussed it with my previous employer.
I was on the call with the HR department from my old company yesterday. They told me that I should have sent them an email before leaving the company, informing them that I don't need the payout and would like to carry over the rest of the holidays to my new company. Also, I am told that it is too late to send an email and it won't work now (I have never heard of such a rule).
The Urlaubsbescheinigung is a bit vague. The exact sentence in there is as follows: "Der Jahresurlaubsanspruch im laufenden Kalendarjahr beträgt 30 Tage; davon wurden von uns (anteilig) 30 Urlaubstage gewährt bzw. abgegolten."
I've given this letter to my current HR and she infromed me that I still have 8 holidays left this year and I should take them before the year ends.
I received my salary slip from my new company today, and I can see that I still have 8 holidays remaning this year.