Sequence of tenses

"She said it was similar to the one she has."

Why "have" in this sentence is in present tense? Is "had" also possible in this case? If yes - is there a difference in meaning?

had is possible and is the norm.  With had you keep all the verbs in the past point of view.
However, you can use the present tense (but you don't have to), even when the preceding verb is in the past, if the situation still exists.  So has is used to indicate that she still has it.

CJ

The present tense makes perfect sense to me, which always makes me think I'm missing something.

I told my sister that I was thinking about getting a Pontiac Vibe and asked what she thought. She said it was similar to the car she has.

I asked Mary whether she thought this plasma TV was a good deal. She thought so, and said it was similar to the one she has.

Is the suggestion that if we want to keep it "has," the "was" should be an "is"? She said it IS similar to the one she HAS?

If the word "had" had been used, that would have suggested that she no longer possesses "it."

No, not at all.  It that were true, then

She said it was similar to the one she [owns/has].

would also mean that it used to be (was) similar but now it's different.  This is clearly not the intended meaning.

Backshifting is automatically understood as a convention in which the literal tense in the backshifted clause is not necessarily the true tense/time of the situation.

If I now say, for example,

In answering a post, I said that backshift was understood as a convention ...,

it does not suggest that it is no longer understood that way.

Similarly,  She said it was similar to the one she owned
does not suggest that she no longer owns it.  It is a possible reading, of course, but not necessarily the first interpretation that comes to the mind of the listener.  used to own would help guide the listener to the "no longer owns it" choice -- if it were important to do so.

CJ

Yes; I understand it in the same way as CJ:

1. "She said it was similar to the X she had."
— Always correct, regardless of whether she still has the X.

2. "She said it was similar to the X she has."
— Correct if she still has the X.

3. "She said it's similar to the X she had."
— Correct if she no longer has the X.

4. "She said it's similar to the X she has."
— Correct if she still has the X.

I've noticed that native speakers will happily use the #1 format, while they're speaking without paying conscious attention to their grammar; but once they notice the apparent anomaly, doubt will set in.

MrP