Learn Russian: У него собственная машина.

The word сам means "oneself" (as in "he fixed it himself"). To describe an object as "one's own", you cannot use сам. Instead, you must use a possessive word like свой or, as shown in this sentence, the emphatic adjective собственный (own/personal).

Could I just say У него своя машина instead? What's the difference?

Yes, you can! У него своя машина is very common and correct. However, using собственная (which comes from the word for "property") adds extra emphasis. It strongly underlines that the car is his personal property—not a company car, a rental, or borrowed from a friend.

Why is the word есть (has) omitted from this sentence?

In Russian "to have" constructions, we use есть to state the mere existence of something (У него есть машина - He has a car). But when we add a descriptive adjective and want to focus on that specific quality (the fact that it is his own car), есть is usually dropped so the emphasis lands entirely on the adjective.

Why is it у него instead of у его?

Whenever a third-person pronoun like он (he) or оно (it) directly follows a preposition like у, Russian adds an н- to the start of the pronoun to make it flow better. Because of this rule, the genitive form его becomes него.