Word for "When two (or more) people are chasing after the same thing"?

Is there a general term in English that describes these collisions?

Contention

Example: There is much contention for parking spaces.

Is there a more formal term for when one person calls "shotgun" for these collisions (i.e. calls dibs, or claims ownership)?

Win

Example: The blue car won the space.

I am going to assume you are looking for a noun that refers to the situation you describe, and another noun for the action of claiming priority in that situation.

I am going to say that there is no such noun in general English. In parallel computing the analogous thing can happen with different processes, and then it is called a race condition. Occasionally that term will be used metaphorically for the kind of "who claims the ticket" scenario you describe in your comment, but most people wouldn't be familiar with the concept. And also it wouldn't feel natural at all to use it with either a love interest or a parking space (two concepts I don't usually lump together).

I think if the contest really just goes to whoever gets there first, you can say that it is a race for X. In the case of a love interest, I imagine getting there first isn't enough, and you could say a competition for X's affections (although that sounds a little old-fashioned).

I think the English term for calling dibs is calling dibs, or just calling something. You can also say that someone "lays a claim", "stakes a claim", or just "claims" something.

I'd say they're vying.

Vie: Compete eagerly with someone in order to do or achieve something.

Example: the athletes were vying for a place in the British team. [Lexico]

Or they're contesting.

Contest: Engage in competition to attain (a position of power)

Example: she declared her intention to contest the presidency. [Lexico]

The competition could also be referred to as cut-throat or dog-eat-dog.

It looks like you're asking for competitions where people are after the same thing, where they cannot all have it.

To describe a competition where two (or more) people cannot all successfully claim the prize, we could use zero-sum game

a situation in which one person or group can win something only by causing another person or group to lose it

(Merriam Webster)

When two or more people eagerly chase after the same thing in a competitive manner, they’re scrambling for it.

Definition: * to struggle eagerly or unceremoniously for possession of something*

Source: Merriam-Webster

Is there a more formal term for when one person calls "shotgun" for these collisions (i.e. calls dibs, or claims ownership)?

In formal language, it's to stake a claim.

[Merriam-Webster, from stake]
stake a claim
: to assert a title or right to something by or as if by placing stakes usually to satisfy a legal requirement

In other words, you are staking the claim that the baseball, parking spot, or love interest belongs to you and nobody else. You are asserting your right to it.

Such people are called rivals. The word is ultimately from Latin rīvālis, "one who uses a stream in common with another" (rivus is stream, whence English river).