Do Unis have a Quota for types of students they accept?

Universities don’t have your ethnicity data yet. This is only released after you start university for use in monitoring stats.
There’s no quotas based on nationality/region (although Scottish universities do have caps on Scottish and rUK entrant numbers due to the way they’re funded). Some universities will take into account university participation and deprivation scores linked to your home postcode. Others will compare your academic achievement to your school averages to identify applicants who are excelling in weaker schools.
Does Durham just look at lnat score or do they do some assessment of the written essay component (assuming that’s still part of the test)?

Ooh that's true, I was thinking that they'd have no clue what ethnicity I am (apart from the big giveaway in my name, and when they met me during the interview)

The LNAT exam is composed of the MCQ section and an essay section so yes, they are said to take that into account. I've heard that both Durham and UCL care about the essay, but 🤣 I don't think UCL even looked at the rest of my application when they saw my poor LNAT score.

OP, both Cambridge and Durham, two universities with competitive entrance standards, have assessed YOU as an excellent candidate. They have based their assessments on YOUR record and YOUR potential. Your nationality, ethnicity, and sex have had nothing to do with the decisions. Congratulations on your offers from two of the UK's leading universities, and good luck with your studies.

When I got my Oxford offer I wondered a bit if it was just because I’m female and working class/state school. Probably just imposter syndrome/disbelief at actually getting into the unis you’ve dreamed about for years. They accepted you because of your application/interview performance, not to fill a quota

Imposter Syndrome is, of course, a curable condition.
It may be that the only people who never experience Imposter Syndrome are people who are in fact imposters. Confident bluffers with little real ability may experience little self doubt.
Or perhaps a faker's knowledge that he or she is a fake is what drives the faker to carry on faking.
People such as Boris Johnson and Donald Trump could satisfy either of those descriptions.
OP, please trust the judgment of the academics at the good universities which have invited you to join them.

Who told you it was because you 'are from Ireland'?
Get this in writing - an email - and then talk to UCAS please.

Oh sorry I must've not been clear. 😭 My dad and my older brother said that they have quotas, hence why I got an offer.
They said that unis must accept "this many people" from different regions/countries and that they want to promote diversity-- indicating that I only received offers because of my nationality/ethnicity. I guess it's not true though.

Oh sorry I must've not been clear. 😭 My dad and my older brother said that they have quotas, hence why I got an offer.
They said that unis must accept "this many people" from different regions/countries and that they want to promote diversity-- indicating that I only received offers because of my nationality/ethnicity. I guess it's not true though.

Just one quick clarification, they almost never have quotas with one main exception: there is a quota on how many home / international students can do Medicine / Dentistry. This has nothing to do with any sort of diversity though and beyond the fee status, everything is treated equally. The most recent data I could find was that 7.5% of total intake could be international (at least in England and Wales) with a limit on the total number of students also.
This is unlikely to affect you in your case but I thought it would be worthwhile to mention it.

You were clear. McGinger appears to have overlooked part of your opening post.
No need to guess, your family members are mistaken. There is no need to get anything in writing or talk to UCAS, because no quota has been applied.