Perplexed if the Battle Axe Proficiency Bonus applies to the swords/rapiers indicated OR is proficiency in the weapon type what was intended

The latter

When you hit a giant with it, the giant takes an extra 2d6 damage of the weapon's type and must succeed on a DC 15 Strength saving throw or fall prone.

Against a Giant, a DC 15 Strength saving throw is a wasted.  DC needs to be much higher otherwise that is a wasted ability of this weapon.

Or change it to a DEX save.

Every time you hit a giant with this weapon they have the make the save, you don't expend anything to do it and there is no consequence to using this ability. It's fine. It's a bonus feature to the main attraction which is free extra damage. It shouldn't be a hard save to make otherwise this item would be a lot more powerful than rare. 

Quote from Andre109 >>

Every time you hit a giant with this weapon they have the make the save, you don't expend anything to do it and there is no consequence to using this ability. It's fine. It's a bonus feature to the main attraction which is free extra damage. It shouldn't be a hard save to make otherwise this item would be a lot more powerful than rare. 

Example a Fire Giant has +7 to Str saves min, a Storm Giant +9....not going to be hard for him to make DC 15, if you want to keep the DC 15 it should be at disadvantage.

I never said it cost anything and if it's a 'giant slayer', it should be powerful.  Using this weapon for situational enemy is costing you an opportunity for wielding another weaon.

Pretty weird there isn't a Giant Slayer hammer of some sort considering Norse mythology.

A DC 15 Strength saving throw would indeed be easy to succeed for a giant, but remember there is a 2d6 that is also applied, making it be with disadvantage would be too powerful for a rare item. if it was a very rare item, what you suggest wouldn't be too powerful.

dude you clearly don't understand balance at all

The prone effect is always on. The only thing it needs to do is hit. And you use high strength giants as your example.
Maybe stop acting like your inexperienced opinion is fact.

The DC is perfectly fine for an item that works on more than just high strength giants and lacks attunement.

The item specifically states, "When you hit a giant with it, the giant takes an extra 2d6 damage of the weapon's type and must succeed on a DC 15 Strength saving throw or fall prone." The effect isn't "always on." It only works on giants. And just going through, all giant type creatures in the basic rules...

Cloud Giant: Strength 27 (+8)

Cyclops: Strength 22 (+6)

Ettin: Strength 21 (+5)

Fire Giant: Strength 25 (+7)

Frost Giant: Strength 23 (+6)

Hill Giant: Strength 21 (+5)

Ogre: Strength 19 (+4)

Oni: Strength 19 (+4)

Stone Giant: Strength 23 (+6)

Storm Giant: Strength 29 (+9)

Troll: Strength 19 (+4)

There are 11, and 8 of which are more likely to succeed than not. 

Yes, this is always on as it doesn't require any activation or even resource expenditures. The creature just has match the criteria of being a giant type.

Just look at it this way. Storm Giant's are completely immune to the prone portion of this as they are proficient in STR Saves and have a high enough bonus. They are the only ones who can auto pass this check. Even if you needed a nat 1 on the save to knock it down, that's still a 5% chance for you to knock it prone (Which is going to give both you and your team mates adv on your attacks) with no resource expenditure. If you had a weapon that was a 5% - 50% (depending on the giant) to grant advantage on any melee attacks before the beginning of the creatures next turn, that would be considered quite good. That is before you even consider the extra damage that will also double on a crit (Which you have double the chance to do once it's prone). Keep in mind, this is EVERY attack that hits they have to make this safe. This is not an "I Win" button against giants but it shouldn't be.

This is a rare +1 item enchantment that could start showing up in mid level play. It is VERY good if there are any shenanigans going on in your campaign with giant types. It's decent if there isn't as it's just a standard +1 weapon at that point.

Could you find or make a "Giant Slayer" +1 or +2?

I agree with you on just about everything as far as the weapon itself is concerned, but I want to point out that a Storm Giant automatically succeeds no matter what, as there are no critical failures or critical successes in saving throws and ability checks.

There is! The hammer of thunderbolts!

I don't get the argument about this weapon.

At worst it is a +1 with no attunement.

Against a giant it is a +1 Flametounge with a knockdown chance and no attunement.

All as a rare item.

Why only edged weapons? Why not Maces, Mauls and Clubs? Is there any specific reason?

There's no reason the DM couldn't adapt blunt weapons for it, whether that's a custom item or just re-skinned, it's all adaptable.

"the giant takes an extra 2d6 damage of the weapon's type"

Shouldn't the dmg be +2 dice of the weapon type used instead of d6? Example a long sword is d8 so bonus should be +2d8. Otherwise the sentence could stop at extra 2d6 damage.

I think by “weapon type”damage,  they mean slashing or piercing.

CAUTION:  the giant-slayer weapons aren't correct in D&D Beyond inventory.   they are only displaying and including the damage di(c)e "of the weapon's type".  they aren't showing or including the "extra 2d6" damage in the rolls from the character sheet when hitting a creature of giant type

https://imgur.com/gallery/giant-slayer-weapons-d-d-beyond-ZZYdjQ0