In short, it is a Bengali (not Hindi or Sanskrit) simple present 3rd person non-honorific form equivalent to Hindi singular गाए (sings), plural गाएँ (sing), and Sanskrit गायति/गायन्ति|
In more detail, the form (गाहे/গাহে) specifically belongs to an older literary standard of Bengali, called "sadhu bhasha". In modern standard Bengali (cholito bhasha) the form is gay (गाय/গায় - one syllable). Indian national anthem is formally in Bengali, but the vocabulary is chosen in a way (mostly proper nouns and Sanskritic nouns and adjectives with zero-ending) such that it is understood by almost any educated Indian without knowing Bengali. There are just a handful of words which bear any grammatical marking - "gahe" is one of them. The others are "name" (নামে/नामे - in the name), jage (জাগে/जागे - wake(s) up) and mage (মাগে/मागे - ask(s) for) - all readily understood by modern Indo-Aryan speakers, even if their languages may use a slightly different form. There is also the form sindhu/সিন্ধু/सिंधु - the river Indus or the province of Sindh - again readily understood by all, even if their language-specific form is slightly different.
