Holiday is regulated by the Bundesurlaubsgesetz in Germany. It explains how many minimum days there are, and how the days you are owed are calculated.
The following parts are relevant. All translations and emphases are mine.
§ 3 Dauer des Urlaubs
(1) Der Urlaub beträgt jährlich mindestens 24
(2) Als Werktage gelten alle Kalendertage, die nicht Sonn-
oder gesetzliche Feiertage sind.
This says you are entitled to at least 24 working days of holiday. Many companies do more, with the generally expected maximum being 30 working days.
§ 5 Teilurlaub
(1) Anspruch auf ein Zwölftel des Jahresurlaubs für
jeden vollen Monat des Bestehens des Arbeitsverhältnisses hat der
für Zeiten eines Kalenderjahrs, für die er wegen Nichterfüllung der Wartezeit in diesem Kalenderjahr keinen vollen Urlaubsanspruch erwirbt;
wenn er vor erfüllter Wartezeit aus dem Arbeitsverhältnis ausscheidet;
wenn er nach erfüllter Wartezeit in der ersten Hälfte eines Kalenderjahrs aus dem Arbeitsverhältnis ausscheidet.
(2) Bruchteile von Urlaubstagen, die mindestens einen halben Tag
ergeben, sind auf volle Urlaubstage aufzurunden.
This regulates how days are calculated if you work for less than an entire calendar year. You are entitled to one twelfth of your overall yearly holiday for every full month (**not calendar month) you are working, after having worked the first six month of your employment. If your employment has been shorter than 6 months, you are entitled to the holiday based on full months. Your partial holiday is rounded up to full days if the partial day is at least a half-day.
So your contract says 27 days per year. That's
2.25 days per month. For the first six months of your employment (which might coincide with your trial period (Probezeit)), you are entitled to those 2.25 days for every full month you have worked. So when you start on the 2nd of January, the period of one month ends on the 1st of February. If you start on the 1st of January, the period of one month ends on the 31st of January.
This means that on the 2nd of February you are legally allowed to take 2 days off (as most companies don't do quarter days). If you don't take that, then on the 2nd of March you have 4.5 days, so now this is rounded up to five days.
If you don't take the holiday during the first six months, you will be allowed to take a longer period than the 23.5 days you have accrued on at the end of June in one go. In theory, you could take three or four weeks in one go from the 2nd of July.
In practice, most companies don't do this waiting time. The companies I have worked at in Germany didn't mind if in the first month you went away for a week if there was a good reason, like having already planned a holiday and telling them during the interview.
In conclusion, I think there is miscommunication going on, or the software they use is flawed. Most companies will use some sort of holiday calculator if they have people working in offices. These programs are unfortunately often not very well designed, as it seems to be a niche market. Therefore it's possible that it just calculates it wrong.
Talk to them again to clarify. Ask to have the contract start on the 1st. If they refuse, ask why. If they say because of the bank holiday, ask why that's a problem. It seems like someone there is incompetent. Of course it could be that they are malicious, but really what do they gain from giving only new employees that start at the beginning of the year two days fewer holiday in the first year? It would make much more sense to simply write into the contract that your holiday is 25 days, and you get two extra days granted from the 2nd year of employment. I've had a clause like this in the past.
According to my research, this means that initially you are not e