How does having a PhD affect your chance of getting into a second PhD UK university in a different field UK university? My friend has taken a unusual route in academia. She first did a bachelor and PhD in electrical engineering, fully funded by her scholarship. However, she told me that she has little interest in engineering, and only did it to please her parents. Her real interest is in art history. Hence, two months after completing her PhD in engineering, she took up a masters in art history in the UK, which is her dream. She did very well in her masters, and started applying for a PhD in the UK earlier this year. As of now, she has received multiple offers to study for a PhD in these universities. But, she has little success in obtaining scholarship for her PhD in art history.
I cannot help but feel a bit 'unfair'? I know many students who have spent a lot more time taking art history classes, such as those who did a bachelor and then a masters, but cannot manage to get into PhD programs. Hence, I have a few questions on the process that the UK universities have in selecting PhD students. Are the UK universities quite relax in choosing PhD students who have their own means of funding (assuming that they can get a supervisor there who is willing to supervise them)? Does the pandemic causes the UK universities to relax their section of incoming PhD students? Does having a PhD in a separate field made my friend easier to be accepted into a PhD in the UK university?
... she has little success in obtaining scholarship for her PhD in art
history... I cannot help but feel a bit 'unfair'?

Overall, I would interpret your friend's lack of success in getting funding for her second PhD as tacit rejections from these universities. An offer of a PhD without any money attached is not really an offer at all (see e.g. Is it worth self-funding a PhD to attend a top 10 university? and especially Fomite's answer to that question for more information on this point).
Note: there is a subtlety here about the type of self-funding. If a student has won an external scholarship or grant, or has some money from their own government (or maybe a national museum or gallery in the case of art history), that student would be very attractive and likely get many offers for a PhD.

Are the UK universities quite relax in choosing PhD students who have
their own means of funding (assuming that they can get a supervisor
there who is willing to supervise them)?

No. If you don't have the potential to do good research then it's likely you won't get an offer to study for a PhD, even if you're as rich as Croesus. Since your friend has been offered PhD positions, then the universities clearly think she is capable, but would rather award their funding to someone else (see my answer to your last question below for why).

Does the pandemic causes the UK universities to relax their section of
incoming PhD students?

It could be that there are fewer applicants for PhD positions, as people may be preferring to go into a more stable career than academia at the moment, or international students may be put off by how the pandemic has been handled in the UK. However, I would expect that there are still more than enough candidates to make the selection competitive and hence no relaxation or lowering of standards on the part of the universities. Furthermore, since PhD funding is decided years in advance, I don't think that we have seen the effects of the pandemic on that yet.

Does having a PhD in a separate field made my friend easier to be
accepted into a PhD in the UK university?

It's possible. Offers for a PhD are generally based on the research potential of the candidate, and obviously someone who already holds a PhD is capable of doing research. However, this clearly counts against them when it comes to funding. Most supervisors/departments would rather fund someone who doesn't have a PhD than someone who does -- who's already &quot;had their turn&quot;, so to speak. This question and its answers discuss why doing two PhDs is generally not a good idea: Is doing two PhDs a good path?.
A final point to consider is that funding in art history is likely far scarcer than in engineering, especially in the UK where the government has placed heavy emphasis on funding STEM research over humanities in recent years.