Your second sentence isn't correct and would not sound natural to a native speaker. Instead, here are some things you could say:
- I have owned that car for ten years.
- It has been ten years since I bought that car.
You are also correct that I bought that car ten years ago is correct.
You are getting confused because there are many different ways that the word since can be used, and you are combining two of them.
Since - Ongoing Events
The word since in English can be used in a few different ways. In one common usage, it connects a present event with a past time to show that something started in the past but is continuing to the present.
Here are some examples with since used like this:
I have been cooking since 10 am. - means that I started cooking at 10 am and am still cooking now.
She has given me gifts on my birthday since we were young. - means that someone started giving me gits when we were young, and still does it.
You have been going to this school since September. - means that you started going to the school in September and are still going to that school.
In most cases (not all, but most), the first verb in a sentence structured like this will be in the present perfect tense.
Since - Past Events
You can also flip the sentence and put the time first with the action second by using the construction it has been or it had been. If you do this, it shows that time has passed after the action happened.
It has been 2 years since I saw my aunt. - this means that I last saw my aunt 2 years ago.
It had been only 10 minutes since the child asked for attention. - this means that the child asked for attention 10 minutes ago.
It has been 10 years since I bought that car - I bought that car 10 years ago.
In informal English, people often use a contraction here and will say "It's been" instead of "It has been."
Why your sentence doesn't work
You are trying to use the second type of sentence - describing how long ago an event happened - but using the pattern of the first type. However, the first type only works if the event is still going on, which doesn't make sense when talking about buying a car.
If you say "I have been buying that car [present tense] since 10 years ago" it implies that you have been in the process of buying the car for the past 10 years, which isn't what you mean. Buying a car is not usually an ongoing action or event. It's a past event, so you need to use the second type of sentence.
