Why isn't high functioning autism recognized by the DSM-5 or the ICD-10? I was looking at this https://en.m.wikipedia.org/wiki/High-functioning_autism and it said that. Aspergers is also closely related to high functioning autism but it is also not recognized anymore. Does this mean that neither of them are a "thing" anymore? Would diagnosing someone with either of these labels be inaccurate?
Asperger's being "subsumed" in DSM-5 is not the whole story. There's indeed a new DSM-5 coding for the severity of the disorder. But applying the DSM-5 actual clinical criteria results in fewer people being diagnosed with autistic spectrum disorders, and the reduction is more significant for the mild versions (such as Asperger's):

A 2014 mini-review looking at 12 papers concluded:


  The application of DSM-5 diagnostic criteria resulted in an approximately one third reduction in Autism Spectrum Disorders. The reduction was approximately two thirds for mild forms of Autism. 


So yeah, the DSM-5 take (intentional or not) is that a lot of "Aspies" don't have a level of dysfunction that raises to the (DSM-5) bar of a mental *disorder *.

However in an astute form of politicking, people with an existing Asperger's diagnosis were "grandfathered in" by the DSM-5. From the same review:


  The American Psychiatric Association partially acknowledged
  the exclusion of mild forms of ASD when the added
  rider to the diagnostic criteria was that “Individuals with a
  well-established DSM-IV diagnosis of autistic disorder,
  Asperger’s disorder, or pervasive developmental disorder not
  otherwise specified should be given the diagnosis of autism
  spectrum disorder”, but this approach to diagnosis of autismis
  strange. It enables individuals who previously were diagnosed
  with ASD to continue to be eligible for services, but results in
  two odd consequences. First, consider two individuals with
  identical symptoms who meet DSM-IV criteria for ASD, one
  of whom was diagnosed before the publication of DSM5 and
  one diagnosed after: The first will be diagnosed with ASD and
  eligible for services, but the second will not. The second
  consequence is that the meaning of the DSM5 term ASD is
  obfuscated as it refers to different entities using the same term.


And a 2016 meta-analysis confirms the effect of stricter criteria (using twice as many studies as the 2014 review):


  Our meta-analysis measured the percentage of people with an ASD diagnosis from the DSM-IV or DSM-IV-TR obtaining an ASD diagnosis from the DSM-5. All 24 studies identified indicated a reduction in the number of people with a DSM-IV or DSM-IV-TR ASD diagnosis being eligible for a DSM-5 ASD diagnosis, a 35 and 37 % reduction respectfully. Our meta-analysis also showed that those with a DSM-IV-TR or DSM-IV diagnosis of PDD-NOS were the least likely to qualify for a DSM-5 ASD diagnosis at 26 and 47 % followed by 55 and 19 %, respectfully of people with Asperger syndrome, and 80 and 87 %, respectfully of people with an autism diagnosis.


A more recent (2017) study on Australian children (where the use of DSM is mandated by certain government programs) found that DSM-5 did seem to stop to the previously increasing trend in rising autism diagnoses:


  De-identified data for 32,199 children aged under 7 years between 2010 and 2015 was utilised. [...] The incidence of autism increased from 2010 to 2013, and then plateaued to 2015. A significant trend-relative reduction in the number of children registered to receive autism-specific funding was evident post 2013, suggesting the more stringent DSM-5 criteria may have curbed the trend of increasing diagnoses over time.




As for why the DSM-5 removed the separate criteria (including Asperger's), quoting from the last paper:


  These previously distinct diagnostic
  categories were removed in light of empirical data
  that questioned the reliability and validity of these subtypes
  (Daniels et al. 2011; Lord 2012; Macintosh and Dissanayake
  2004; Mahjouri and Lord 2012).


Whether this was the right thing to do or not... it depends who you ask. Asperger's diagnosis definitely has its supporters, at least one of whom even predicted its return in the next version of the DSM.