Aside from the fact that "bike's time" and "Anna's time" don't make even sense in English (except for a very small set of contexts), I can't think of many scenarios where you'd use "possessive" time except for the following. And generally
時間 refers to the amount of time or the specific time of something, so it wouldn't always be interchangeable with
とき without a slight change in meaning.
Representing an "era" or (subjectively) long period of time with
- 子供のとき → When I was a child
- 留学のとき → When I studied abroad
- 留学の時間 → The amount of time I studied abroad
When using a noun from which a similar/exact verb could reasonably be inferred, meaning "It's time to 〜", "〜 time", or "when (I) 〜"
- ダンスのとき・時間だ！ ←→ ダンスする／踊るとき・時間だ！ ←→ It's time to dance! / It's dance time!
- ○ 食事のとき ←→ 食事をする／食べるとき ←→ When eating
- ○ 散歩のとき ←→ 散歩する／歩くとき ←→ When I (take a) walk
- ? バイクのとき → Doesn't make sense because バイクする isn't a verb, and it can't be reasonably inferred what's happening with the バイク; are you riding it? Fixing it? Buying it? In such case, you need to specify unless context or familiarity with the listeners would already afford that knowledge (which is to say probably not often)
- ○ バイクに乗るとき → When I ride my motorcycle
- ○ バイクの修理をするとき → When I'm repairing my motorcycle
- × アンナのとき → Doesn't make sense at all. Again there is no default verb one could infer from アンナ, so you have to specify what you mean
- ○ アンナといっしょにいるとき → When I'm with Anna
The only time I could see "bike's time" making sense is if you meant something like "the time displayed on the motorcycle's dashboard clock", in which case you'd be better off saying something like
バイクの内蔵時計の表示する時刻. Similarly, I can only see "Anna's time" making sense in the context of some kind of race, and you're referring specifically to the time it took her to finish (