Can I fly to Germany from India?

The rule has always been that people with a residency permit can enter Germany. This is AFAIK pretty standard as far as it comes to Covid-related entry requirements around the world.

Deutsche Staatsangehörige sind von den Einschränkungen nicht betroffen.

Auch Unionsbürger und Staatsangehörige des Vereinigten Königreichs, Liechtensteins, der Schweiz, Norwegens und Islands und ihre Familienangehörigen der Kernfamilie (Ehepartner, minderjährige ledige Kinder, Eltern minderjähriger Kinder) sind von den Einreisebeschränkungen ausgenommen.

Das gleiche gilt für Drittstaatsangehörige mit einem bestehenden längerfristigen Aufenthaltsrecht in einem EU- oder Schengenstaat oder dem Vereinigten Königreich (Aufenthaltstitel oder längerfristiges Visum) und ihre Familienangehörigen der Kernfamilie.

(from https://www.bmi.bund.de/SharedDocs/faqs/DE/themen/bevoelkerungsschutz/coronavirus/coronavirus-faqs.html => "Welche Reisebeschränkungen gibt es im außereuropäischen Luft- und Seeverkehr?")

Your Fiktionsbescheinigung is probably due to § 81 Abs. 4 AufenthG, which means it is essentially an Aufenthaltstitel.

So once you make it to Germany on a plane, you should be able to enter. The more difficult problem may be getting the airport personnel in India to allow you to get on the plane, due to the unfamiliar combination of travel restrictions for non-residency permit holders and you only having a Fiktionsbescheinigung. Might be a good idea to check with the airline first, get some written reply and arrive early at the airport for your return trip in case there are problems.

Re. possible quarantine: My impression is that countries that are trying to eradicate Covid-19 have very strict quarantine rules (i.e. quarantine "hotels" that you cannot leave and have to pay for out of your own pocket). Sometimes these rules create a bottle neck because quarantine capacities are limited and so only a certain number of people can enter the country per week (e.g. in Australia, but I am familiar with at least one other country with the same problem)

Quarantine requirements in countries that do not fall into that category are considerably more lenient. Germany currently falls into the category of countries that do not really attempt to bring Covid cases to zero, and quarantine requirements are not very strict. I do not think this will change until January, but you might want to follow the news. And also note that there is a possibility that India might tighten some rules and air travel gets more limited as a result.

P.S. there is also a possibility that new entry requirements are added. One such requirement is already in force (registration at https://www.einreiseanmeldung.de/ ) but it is easy to imagine more might follow. This is also something to watch and keep in mind. E.g. if they, say, introduce a requirement for a negative Covid test before you can board the plane, you would have to prepare accordingly

Another point to keep in mind: A Fiktionsbescheinigung is quite short-term. If somehow you do not make it back to Germany in time (and the probability of something like this happening is much higher than normal at the moment) things might get complicated.

A few points

1. I’ve entered Germany from a risk area twice and never had to answer any questions at immigration at all. I do have a German passport though. Generally it falls to the airlines responsibility to make sure that no one enters a Germany bound plane who doesn’t have the right to enter. Once I got to Germany, control and enforcement was basically non-existent (which was kind of scary).
2. Rules for entry and quarantine are confusing and in flux. Rules for quarantine and local travel are also made by they states, not the federal government. So you should study the current rules of Hessen (Frankfurt) and Bavaria (Wuerzburg). Sometimes the rules even vary from county to county.
3. Jan 15 is a long time away. The situation in Germany has gotten a lot worse since the summer and as a result the restrictions have been tightened up in November significantly. At this point no one knows what the rules in January will be. Rules can change very quickly.
4. Example: Bavaria updated their quarantine and travel rules last on Nov 5 going into effect Nov 9. These rules were supposed to apply until Nov 27 but no new ones have emerged yet and no one knows what’s going to happen even next week.

Conclusion: the Covid pandemic is a global challenge. It’s impossible to predict if or under what conditions you will be able to enter Germany on Jan 15. If you decide to travel, you have to accept the risk to be stuck in India for a while.