I am making a Pact of the Blade warlock/Assassin rogue. I have the Improved Pact Weapon eldritch invocation, so I can make a longbow pact weapon while at range and a melee pact weapon when not.

When I create a pact weapon, can I give it different weapon properties? Or do they stay the same as those listed in the Weapons table?

For example, a regular battleaxe is 1d8 and versatile (1d10). Could I make a battle axe with finesse?

The properties will be the same as the normal weapon

You can choose the form that this melee weapon takes each time you create it (see chapter 5 for weapon options). (PHB 107)

Note that the ability explicitly tells you where you look for your options. If the weapon you want isn't listed in chapter 5, you can't choose it. A battleaxe with finesse is a different weapon from the battleaxe listed in chapter 5 and thus is not an available option.

Pact of the Blade does not give the ability to mix-and-match weapon features

You can use your action to create a pact weapon in your empty hand. You can choose the form that this melee weapon takes each time you create it (see chapter 5 for weapon options). You are proficient with it while you wield it. This weapon counts as magical for the purpose of overcoming resistance and immunity to nonmagical attacks and damage.

—Pact Blade, Player's Handbook, pg. 107

The fact that the paragraph explicitly references chapter 5 as a reference for weapon options is pretty clear synecdoche for how this feature is meant to behave: you are meant to choose one of the weapons in that chapter to determine what your pact weapon looks like and has the stats for.

Creating custom weapons (i.e. a Battleaxe with the Finesse property) is instead in the purview of the DM, and up to their discretion. If your DM decides they want to issue such a weapon, they're certainly free to do so, and if they then go on to decide that they want to allow a Pact of the Blade warlock to summon their Pact weapon in the form of this weapon, they would have to house rule the normal rules for Pact of the Blade, as neither of these things are the Rules as Written for the Pact of the Blade feature.

Pact of the Blade only lets you choose from the PHB weapon options, and that's important for balance reasons.

The only guidance the pact blade feature gives "(see chapter 5 for weapon options)." This strongly suggests that the weapon options are precisely the ones in chapter 5, which don't let you mix and match. If that rule was intended to give you more options, it would specifically say so.

Separately, however, the weapon choices from chapter 5, and the features associated with each weapon, are an important part of how the game is balanced in the first place. Weapons that do higher damage tend to have properties that impose limitations, such as the heavy or two-handed properties. The properties light, versatile, finesse, and thrown are all "good" properties, and if a particular character design can load them onto the heaviest-damaging weapons, that's game-breaking in a couple ways -- it means that everyone "should" take that character design, and it removes the advantages from taking other character builds. As designed, part of the reason a character would lean into Strength in the first place is to be able to benefit from the heaviest, most damaging weapons, and it's important to preserve that aspect to reward players that lean into that aspect of their character.

(All of this is completely separate from the "realism" aspect -- that using a battleaxe doesn't seem likely to involve finesse in the dictionary sense, which Google suggests is "intricate and refined delicacy.")