the badge is their traditional one since forever. for instance, they used this for their 75th anniversary in 1974:
https://hu.museum-digital.org/index.php?t=objekt&oges=746175
(fun fact: the above, official badge has a blatant typo, it says ferenvárosi instead of ferenCvárosi. hungarian quality control in the mid '70s.)
the new, current logo, introduced last year has the same design elements with the same number of stripes (5 green stripes with the white ones in between), the same green badge with the 3 'E' letters meaning "erkölcs, erő, egyetértés" ("morality, strength, harmony/agreement"), the name of the club, and the abbreviated "budapest, ix. ker" (9th district) reference to the district they represent.
so the club is not related to bayern in any ways. the sponsorship is an unrelated, but interesting subject: the prime minister is a huge football/sports fan, so bigger multinational companies will have some privileges if they sponsor football clubs. telekom heavily invested in hungarian sport, and to everyone's surprise, they are winning most telecommunication related public procurement tenders. so there's that too.