One has to shift the Forbidden Planet timeline by a century and a half at least to make it fully compatible with the Trek one. Regardless, NCC-57D there could well be an early Federation vessel from the 22nd century, still without them newfangled transporters or long range subspace transceiver sets.
Possibly not a Starfleet vessel but a civilian police cruiser? This would explain the uniforms without the arrowheads, and the fact they serve the United (Federation of) Planets rather than its specific Starfleet. Their humble ship is tasked with a simple "delayed SAR" errand, something Starfleet generally spits upon ("The Cage"). And they fumble that big time, mistaking Altair IV for a goner just because there's a big flash.
A few decades later, surplus ships of that now outdated type are of course sold to foreign interests, including certain felinoids... NCC-57D remains in service till the 2370s, though.
Timo Saloniemi