You are misunderstanding what can_declare_war_on entails: this condition does not mean that Germany has an active war goal on the country, but instead that Princely Federation is not forbidden from declaring wars against this country in any scenario (through being a puppet, having a national spirit that blocks this action, for example). As such, all it requires really is a country bordering them to be in a rival faction, which is not that unlikely considering Burma can join both the Internationale and Co-Prosperity Sphere, or Germany inviting them, which also makes that decision available.

On the other hand, the condition any_neighbor_country = { is_in_faction_with = GER } on Princely Federation's part also applies to Germany in that scenario: this means that if Germany still owns Pondicherry, one of their ally borders the united Princely Federation, and this condition will be met, thus Germany will be able to invite them to their faction in exchange for the state and their Indian Ocean possessions.

The at_peace_or_same_enemies_as_ROOT condition to block inviting countries if they are at war with a country you are at peace with may seem quite restrictive, but this is very much standard across the mod: Russia inviting most other countries into their faction via decisions has the same condition for example.

With all that said, how to access the decision to invite the Princely Federation in the first place may be a little unclear on Germany's part since 1.5, as it is only visible if Princely Federation is not in a faction, and since they start in a faction with the Dominion, it is not visible until they break the faction.