Do I need digital registration (einreiseanmeldung) in Germany if I have an onward flight from there (different PNR) and do not exit the airport?

Consider that for the first airline, your final destination is Germany, and that they are likely to check you have all the paperwork required to enter the country. They are likely not to care about you having an ongoing flight (which you may miss or have any other issue which could cause you to have to stay in Germany).

Airlines are often more strict than the authorities (which in this case are unlikely to require the registration), because they can incur fines if they let through someone without the proper paperwork.

Since both Sweden and Germany are within the Schengen Area, you won’t go through immigration (but you’ll go through exit passport control in any case). You may however have to exit to secure area (“airside”) if you have checked luggage (which you would need to reclaim and then drop at a check in desk), or if you can’t do online/mobile check-in and the second airline doesn’t have transit desks you can reach on you mr way between the two flights.

Yes, just changing planes in Frankfurt counts as a transit. Even transit on land (e.g. by car, train or bus) is exempt from the requirement to register. So no, you do not have to register.

You will find this confirmed on the very first page of the entry registration form:

When do I not need to provide my details?

• If you ... are only passing through Germany.

The wording of the regulation (in its official English translation):

Section 3 (obligation to register) ... does not apply to persons who ... are only passing through the Federal Republic of Germany and will be leaving the country by the quickest route to complete their transit.

So as long as you are not planning an unnecessarily long stay in Frankfurt, e.g. to take a trip into the city or something like that, and are just entering to wait for your onward flight, you clearly fall into this exception from the obligation to register.

The last time I travelled from Sweden to Germany, about 5 weeks ago, I was asked at check-in wether I was going to stay in Germany or was just entering for the purpose of transit. Since I was going to stay, they wanted to see my entry registration. I would very much assume that also the check-in staff at the airport in Stockholm is aware of the difference. If they for some reason are not and demand to see an entry registration, you can always do that at the airport. It only takes a few minutes, but you do in that case indeed need to 'invent' an address, at which you are going stay in Germany.

Does this qualify as a transit?

Yes. You don't need to fill out a registration but things will be a lot easier if you do. Lufthansa in Stockholm will need to verify that you meet all the entry requirements for Germany. The fact that you have an onward ticket for the same day doesn't matter to them. If you need an address, just use the airport or a airport hotel. There are two Hilton's right at the train station.

If you don't have the form, you need to sweet-talk the check in agent to accept your Air India ticket as proof of departure. This may or may not work and Lufthansa can deny you boarding.

Would I have to face immigration at Frankfurt in this case?

No. It's a Schengen to Schengen flight so there are no passport controls on either end. You need to figure out how you can check in to the Air India flight. If you can do it online, you can simply stay in the secure area. If you need to see a check in agent (for document checks, for example), you can try a customer service desk (if you can find one) or doing it at the gate. The safest bet (if you have the time) is to leave the secure area and go to the regular Air India check in counter. Clearing security in Frankfurt is typically no fun, so you want to budget at least an hour for this.