Learn English – get (for) you

Of the past is an adjectival phrase, but in the past is an adverbial phrase. So your first sentence, if you substituted one phrase for another, would have different meanings. To say that "phones in the past were landlines", which contains the omitted verb "used" before "in the past" does not suggest that modern phones are landlines, but "phones of the past were landlines" says that those which have survived in someone's basement still are landlines.

This is implicit in your second sentence fragment (which is not a complete sentence), since presumably the rest of the sentence compares transportation in the past to transportation today, and either discusses the differences between them or establishes similarities - but either way the two classes of transportation are assumed to be different. If they were not different there would be no point in comparing them. But "transportation of the past", such as horse-drawn carriages or steam-driven locomotives, has not changed today, although it may not be used much.

You belong with me.

This refers to someone being in the right place (generally because the speaker and the person being spoken to are in love) when they are with the speaker. This "being together" is of course not necessarily a matter of being physically near to each other, and instead it can refer to 2 people being in a committed (and ideally monogamous) relationship with each other.

You belong to me.

In this sentence, the speaker is expressing "ownership" of the person being spoken to. It's more one-directional, almost ignoring the other person's opinions on the matter.

I guess if we were to read into this a little further, the first sentence implies a sense of "correctness" in the two people being together, as if the circumstances make them suitable for each other, in addition to their mutual attraction. Call it "destiny", if you want to.

The second is definitely more egocentric, stating that the speaker claims (ideally exclusive) ownership of the person being spoken to, and therefore also claiming the right to govern their actions. But this unbalanced power situation also implies that the speaker may also "own" other people (i.e. lovers), and that the person being spoken to is powerless to change that.