I informed my boss that I have a doctor's appointment at 11 am and 5 pm on different days.
Good, you did the right thing. You informed them early, you did not "fake" ask for permission and then wonder what you do with a reply you don't like. Exactly right.
He responded [...] telling me to record my doctor's appointments as absences on my timesheets
Your boss acted absolutely correct here, too. Doctors appointments are protected, they cannot fire or reprimand you for going, but claiming you were at work is just wrong and would be fraud on multiple levels. Fraud vs your employer, vs the employers accident insurance, vs the tax authorities if you actually commuted... many people get to know whether you were actually at work and the truth is you will not be, that is the whole point of a doctors appointment.
Am I required to record doctor's appointments as absences and what are the consequences for me if I do?
Yes, and they can fire you if you claim you were at work instead. Going to a doctor is protected, lying, falsifying time sheets and claiming you worked during that time is not. This is not your employer making life hard or controlling or micromanaging you, having correct time sheets is a legal requirement for them.
Therefore he's implying I can't attend doctor's appointments during work time
No, they are not. They did not say so or imply so. They said you cannot record your doctors visit as work time, which should be obvious, because it isn't.
I have a feeling he's looking for a way to duck my pay during my notice period. I can offer to make up that time later by working overtime but I don't know if this is even necessary in Germany.
If you go to the doctor and get a sick note, then you will miss the day or more days to come and they will have to pay you (that money will be paid by your health insurance behind the scenes, but you will get a paycheck as if you had been at work).
If you go to the doctor and not get a sick note, then it would be up to you to prove, that there was no way to make that doctors appointment outside of working hours. Since you said you were working remote in a startup, I will assume you have rather flexible working hours. The expectation is that you will make up for the lost time. Either by starting earlier on a day where you have to leave early, or by working later on a day where you had to come in late.
Different rules may apply if you work a job with strict shift requirements and strong labor unions, like a factory worker at an assembly line, where you need to be there when it starts and the whistle blows at 16:30 you cannot work individual overtime. But that is not you. You are flexible. You are expected to schedule your work to accomodate a doctors visit, normally by making up the time on the same day or another day.
They have to accept your doctors visit, they don't have to pay you for not working. Unless you get a sick note.
That is pretty much standard in Germany for the last quarter of a century that I worked jobs with flexible hours.
