I wouldn't say it's outdated when many people use it for ourselves, but there are always going to be limits with these sorts of terms.
Mixed or multiracial is fine as a general population level term with the understanding that it is limited, but individuals can obviously have their own feelings and preferred terms. I use mixed (or Mixed Other on forms) or Mestizo.
I did diversity training recently and the said we should never use mixed race - a complete no no We were told person of colour, diverse racial or enriched racial background.
I would have found that a very difficult diversity training to sit through. Enriched and Diverse feels really patronising. FFS, we're not flour.
What about those of us who are mixed race in part because one of our ancestors was raped? Not something I'd want to call 'enriching' or 'diverse'.
Person of Color originated in the States as a political term of solidarity - people of different races who came together to fight for civil rights. Applying to everyone who looks a certain way has largely stripped the political significance of that term (some argue intentionally) and also has been an issue when certain nationalities are painted as 'PoC' when the nation themselves have people of all colours.
And of course, there are mixed race people that others read as White some or all of the time, some identify as White because of that or just personal preference. That training really oversimplifies being mixed, it kinda feels like they view it as a bad thing that needs to be semantically made better rather than just a neutral concept. I describe how I talk as a 'mixed accent' because that's what it is, it isn't any better or worse, it's just a mix of multiple dialects.
I'm Metis on my mother's side, Mestizo on my father's. Both of those terms just mean mixed but - unlike most colonial racial terms - they'e developed to have their own meanings and cultures differing by group mixes and location. I've identified as mixed, Metis, and Mestizo over the years for various reasons, though I mostly use Mestizo now partially because there is a long tradition of standing up against blood quantum and just the concepts that are kinda baked into things like Dual Heritage or Half Caste where our ratio matter when more often than not, it's been used by powers to divide and reduce us. I am not the ratios of people before me. I don't expect Brits to know that term or to identify me as such by looking at me - part of many Mestizo cultural ideas is how we look like all sorts - but I do use it when it comes up.
BAME - it's better than PoC to me for population level discussions when its understood that it's oversimplified and nearly always needs to be divided down more if we're actually to understand (and I tend to use the Black, Asian, & Marginalized Ethnicities over minority because that tends to be more the issue). Like LGBT, I don't like it being used on individuals - I am bisexual not LGBT, I am Mestizo not BAME, but reading the area I live in described as having a "growing BAME population" doesn't bother me when it's a starting point before moving into or discussing the need for more details. It gets iffy when it's treated as a shield/tickbox like the government or a company having 'BAME representation' like that means something.
I think the Americans say bi-racial and think that mixed-race is offensive even though logically, the words have the same meaning (except that 'bi' is limiting it to just two, so if you correct somebody who said mixed race and insist that bi racial is the word now it makes an assumption)
I never heard biracial or dual heritage used until I lived in the UK. I'm sure it's used in the States, but it's likely more a regional thing (likely the West Coast and/or New York coastal which tends to be where this type of stuff that then comes over tends to be rooted in) rather than an American thing.