Applying to a uni while doing undergraduate at a different one

It's possible to apply elsewhere, but not for Oxford.
Broadly-speaking, Oxford won't consider this unless you are planning to join them for a second undergraduate degree.
https://www.ox.ac.uk/admissions/undergraduate/applying-to-oxford/second-undergraduate-degree
"If you are currently studying at a university (either in the UK or internationally) and are thinking of applying to Oxford to start the first year of an undergraduate course (and you are not planning on completing your first undergraduate degree), please note that normally we will only consider such applications in exceptional circumstances. You should also make very clear in your application why you do not wish to continue on your current course. "

Do you know why this is? This seems a bit silly 😭 and does that mean someone on a gap year is more preferable?

Note this only applies to students at UK universities, they don't have such a policy for international universities in part because the educational systems of some other countries is not comparable to the UK system and so sometimes the only way they can achieve an equivalent level of education for entry to the degree is to complete the first year of a degree in their home country.

It's because they have a policy of not "poaching" students from other universities - they have to work with those other research institutions and if they just took any student who reapplied after starting a degree at another uni that met their requirements otherwise, it would negatively impact those other unis and would not be conducive to a productive collaborative research community.
Cambridge have similar policies but will consider it with "strong support from the current university" (which you won't get after being there for 2 weeks in first year which means you necessarily end up with student finance issues).
Note this only applies to students at UK universities, they don't have such a policy for international universities in part because the educational systems of some other countries is not comparable to the UK system and so sometimes the only way they can achieve an equivalent level of education for entry to the degree is to complete the first year of a degree in their home country.
What this means for you is that you will not normally be accepted if you are currently enrolled at another UK institution. If you have significant extenuating circumstances necessitating you be resident in Oxford or are unable to continue in your current uni (e.g. you require specialist medical treatment only available in the Oxford area, or you have some serious trauma that has occurred and for PTSD reasons you can't continue there or something).
If you want "another bite at the apple" you need to take a gap year. Otherwise plan to complete your gap year elsewhere then look at doing a graduate course at Oxford.

Oxford have now added “and internationally” to the above linked page. I think that’s new for this year.

Oh interesting, that's definitely a shift from before as I understand!
Good to know for future ref