First, a note for those who are not familiar with the terms. In general, double degree or joint degree are degrees awarded by two different universities within a student mobility agreement. They do not refer to the pursue of two independent degrees, as described in this question.
At the Master's level the tradition of double degrees is well established within the EU, and has been going on for more than twenty years, probably 30, since the establishment of the Erasmus programme. Currently, the major EU programme supporting joint degrees is the Erasmus Mundus. The Erasmus+ guide, p. 109, describes the framework of the Erasmus Mundus Joint Master degree. Of course, individual universities can set up joint degrees outside of this programme, especially when one of the partner universities is outside the EU.
At the PhD level things are much less established. As far as I know, the idea of a double PhD degree has emerged only in the last 3-4 years. Before, it was totally unheard of, at least in my country (Italy). Though students are usually encouraged -- and funded -- to spend a period abroad (typically from 6 months to 1 year), this is done within an agreement between two research groups, and not at the university level (though it should be approved by the PhD programme committee).
Usually, to obtain a double PhD degree a student should follow a shared path between two partnering universities. For instance, the two universities might agree that the student shall spend two consecutive years in one university under the supervision of X, and two other consecutive years in the second university under the supervision of Y; that the coursework should be divided equally between the two universities; that the student should defend their thesis in front of a joint committee (or in front of two committees separately); etc.
Since at the PhD level things are much less established with respect to the Master's, I will answer on the basis of my experience: I have a student who is pursuing a double PhD degree; one who initially asked me to do that; and I know a couple of other students from other advisors who are pursuing it.
None of the universities I know of have an explicit double-degree programme for PhDs. In fact, the double degree is usually proposed by a student who thinks, in this way, to strengthen their PhD or to facilitate the immigration in another country. This means that the double degree should be set up on a case-by-case basis, which usually brings about a bureaucratic nightmare, especially for the PhD schools and the advisors. This nightmare comes from the fact that different countries, even within the same region (e.g., EU), have different requirements for what concerns programme duration, funding, coursework, publications and assessment, and finding an agreement is quite complicated.
For instance, if the PhD in one country has a duration of 3 years, and it is funded for that period, who is going to pay for an additional year in a country where the "standard" PhD duration is of 4 years? If there is an ongoing collaboration between two groups it might not be that difficult to find the money, but it could be almost impossible to set up such an agreement out of the box.
As I said, in my experience, the double degree is usually proposed by a student who thinks to strengthen their PhD. I strongly disagree with this premise, and I usually discourage my students from such an idea. A double degree is not stronger than a single one done working hard and with a good publication record. Then, whether it can facilitate or not immigration to another country is questionable: I don't think that, to date, there is sufficient statistics to know this.
As for your last questions:
for an advisor, what is the advantages of having a part-time student?
There is an advantage if there is a collaboration between two groups, otherwise there is really no advantage.
For last, the thesis needs to be approved for both universities... how this happens? I would have to present for both?
These details are really decided on a case-by-case basis, and sometimes the outcome depends on whether one university wants to appear stronger than the other and impose its criteria.