What is it called when there is no personal increase of wealth but GDP increases because of mutual sales? In a hypothetical situation, every person produces some product but instead of consuming it sells it to another person.
This situation would increase the GDP whereas self-consumption doesn't.
Say, person x cleans person y's car and vice versa, and both pays each other some amount Z. Or maybe person x cleans y's car, y cleans a's car, a cleans b's...and so on. Or let's say x sells eggs to y, y sells to x.
Either way, the persons involved got some product or service and is left with no extra money.
This situations increased the GDP, whereas self-consumption doesn't. Is it a flaw? If so, what's it called?
Yes this can be considered flaw of GDP since GDP’s intended purpose is to measure the value of output produced by a nation.  In fact in his report to U.S. Congress Kuznets (who is the person who developed the modern concept of GDP) mentions in a preface that national product (GDP) in his view should encompass all production at its hypothetical market value (not just a production actually sold at market):

If  all commodities produce and all personal services rendered
during the year are added  at  their market value, and  from  the resulting  total  we subtract  the  value  of  that  part  of  the  nation's stock  of  goods  which  was  expended  (both  as  raw  materials  and  as  capital  equipment)  in  producing  this  total,  then  the  remainder  constitutes  the  net  product   of  the  national  economy  during  the  year.  It  is  referred  to  as national  income  produced,  and  may  be  defined  briefly   as  that  part  of  the  economy's  end-product  which  is  attributable  to  the  efforts  of  the individuals  who  comprise a nation. [emphasis is mine]

Kuznets even explicitly mentions house production should be part of this income, but he concluded it’s omission is ‘unavoidable’ due to the fact that it is problematic to measure it:

Services of housewives and other  members of the family.
—The  volume  of services rendered by housewives  and other members  of  the household  toward the satisfaction  of wants must  be imposing indeed,
when totaled for  the  30 million families  comprising  the population  of  this country; and the item is thus large enough to affect materially any  estimate  of  national income. ... It  was considered  best to omit this large group of services from national income, especially
since  no  reliable  basis is  available  for  estimating  their  value. This  omission, unavoidable though it is, lowers the value of national income
measurements as indexes  of  the  nation's  productivity  in  conditions  of  recent  years  when  the  contraction  of  the  market  economy  was   accompanied by an expansion of activity within the family [emphasis is mine].

So already the original creator of GDP arguably considered this omission as a ‘flaw’ but an ‘unavoidable’ flaw.
In addition historically GDP was from its inception used also for welfare comparisons and economists generally agree that home production and self-consumption increases welfare (see England; 1998).
Consequently, it is valid from economic perspective to consider omission of non-market activity (such as home production) as a failure.
This being said  from purely accounting perspective this would not be flaw because GDP is defined as “the total monetary or market value of all the finished goods and services produced within a country's borders in a specific time period” (see here). So from pure accounting/statistical perspective it’s by definition not intended to measure non-market production, but as mentioned above this accounting/statistical definition was a result of expediency and not because it’s intended purpose was to omit non-market activity.
I am not aware of there being any special name within economics for this flaw. Generally, the above would be just called limitation of GDP, but limitation here is not some economic jargon.