No ordinary case exactly covers the Broom of Flying, but the most reasonable approximation is to grant a flying speed.
Is the broom a mount?
By an reasonable interpretation of the Mounted Combat rules, no. To be covered by those rules, a mount must be "A willing creature that is at least one size larger than you" (PHB p.198.) The broom of flying fulfills neither of these conditions. The combat rules also include clauses that clearly cannot apply to the Broom as it is described: the broom cannot Dash, Disengage, or Dodge, since it is not a creature. It has no place in the initiative order. It cannot be independent.
Of course, if the broom specifically stated that it was to be treated as a mount, that would override the general restrictions on what a mount is, but it does not do so.
Is the broom a vehicle?
Probably not, since vehicles in 5th edition seem to never be self-powered. You could make an argument for this, but it wouldn't help you much, since as far as I'm aware there's no generic rule for controlling vehicles in combat.
Is the character granted a flying speed?
Technically no, since other magical items that do this explicitly say so (winged boots, DMG p.214.) I rate this the closest interpretation, however, due to the Special Travel Pace section of the DMG (p.242) which lists "flying speed or with a speed granted by magic, an engine, or a natural force" as equivalent for the purposes of determining long-distance travel pace. The rider of the broom could be considered to have been granted a speed by magic under these rules, since they are traveling using the flying speed of a magic item. Notably, the carpet of flying, which is worded similarly, is listed as an example.
So how do we resolve this in combat?
Despite none of these options strictly applying, I have a hard time seeing where granting a 50 ft flying speed (hover) would give bad results. In practice, this is what I would recommend using.
If you are the DM and you'd like something with more precision under the existing rules, I'd additionally recommend slightly modifying the wording to make it clear to the player how the broom can be used. If you want to treat it as a flying speed, make it explicitly grant that. If you want to treat it as a controlled mount, explicitly state that.
Addendum: Can the broom be remotely controlled?
Not well. The broom can be sent to a particular location, and recalled, but the use of a command word for both tasks indicates that there is no room for interpretation or nuance besides giving it a destination and letting it loose.