There does seem to be a feeling in Northern that the logo has to be some form of N, whereas it generally doesn't seem to be the case at the majority of other TOCs that the logo has to be based on the initial letter.
Going through the list of TOCs:
Abellio Scotrail: Stylised Saltaire
Avanti: Stylised letter A
c2c: none/name
Caledonian Sleeper: Stylised stag
Chiltern: none/name
Cross Country: Stylised X (from abbreviation XC)
EMR: Abbreviation
GTR
{Great Northern: flying triangle
{Thameslink: none/name
{Southern: roundel
{Gatwick Express: Abbreviation
Grand Central: Winged circle with abbreviation
Greater Anglia: none (hare for local lines)
GWR: Abbreviation
Heathrow Express: Stylised X
Hull Trains: Heart
LNER: Abbreviation (/Stylised N)
London Overground: Roundel
Merseyrail: M in a circle
Northern: n in a circle
SouthEastern: Abbreviation
SWR: stylised network map
TfL Rail/Elizabeth Line: Roundel
TPE: 'pole star'
TfW: Stylised T
West Midlands Trains:
{LNWR: Stylised W
{WMR: Diamonds (Transport for West Midlands standard pattern)
In most cases () the 'logo' is either non-existent, or an abbreviation of a name too long to be condensed nicely. In northern's case, the abbreviation does just become N of course
A lot of the name disconnected designs (where they haven't just stylised the name/a letter in it) have got existing heritage or an obvious source: Scotrail, Southern, LO, TfL Rail, & WMR. It's only Caledonian Sleeper, Hull Trains, SWR, TPE (and possibly GN's triangle) that are real 'independent' logos. Notably most of them are also intercity operators where the branding is all the more important - quite what name-disconnected logo you choose to do for a mediocre commuter operation I don't know!