Some cut outs appear to be symbols of the group, such as Christians with a cross cut out. This appears to be a bad idea, ruining the structural integrity of the blade of the weapon. It looks interesting but did it ever exist or serve any purpose?
They certainly existed. This is more common with axes and various polearms than with swords, but there have been swords and daggers with pierced blades. Sometimes the main purpose is decorative, and sometimes the main purpose is to reduce the weight of the weapon.
In principle, cut-outs etc. can be useful for stopping cracks from going across the whole blade. But this might just be an incidental benefit, rather than being a reason for making such weapons.
This appears to be a bad idea, ruining the structural integrity of the blade of the weapon.
As long as there is enough material left, the weapon will be strong enough. Yes, it is a bad idea to remove so much material that the structural integrity is ruined, but typically a lot of material can be removed without making the weapon too weak.
Note how much material has been left on these pierced axe heads:
https://i.pinimg.com/736x/de/ea/1f/deea1fc3d83853c28b9ebb77f0bd8106.jpg
https://i.pinimg.com/736x/7b/43/c6/7b43c66f02abd41c1a2a151288ed855f--battle-axe-the-vikings.jpg
and compare with how much material is present on these non-pierced axes:
In some cases, the pierced axes have more material supporting the blade, and at worst, not much less. So they should be quite adequately strong. Some axe and polearm blades are pierced, but the holes are quite small, and it would make very little difference to the strength:
https://i.imgur.com/WiV1FWw.jpg (the two blades on the left have piercings, and the one on the right has dots - similar marks but not going all the way through the blade)
Other axe and polearm heads are primarily decorative/ritual/ceremonial:
https://i.pinimg.com/564x/49/b2/45/49b245044baaabaa2c3598e0a51e0c62.jpg
https://www.metmuseum.org/art/collection/search/310808 (this example is royal regalia)
Even though these are not primarily intended for fighting, there is still enough iron/steel/bronze supporting the edge so that they should not fail too readily - assuming that they are sharp, they would work as fighting weapons even if that isn't their main function.
Spears can also have decorative cut-outs:
It is less common with swords, and where swords are pierced, it is often small holes in the ricasso, where the blade is quite thick and will easily retain enough strength even with such holes:
Another type of decorative piercing is slots with small balls in the slot, which can roll back and forth along the slot. Sometimes it is said that these function to add weight to the tip of the blade mid-swing, but the mass of the balls is too small to make a significant difference. This feature usually only appears on Indian and Chinese weapons, where it is usually called "tears of the wounded" and "rolling pearls", respectively. Here is an example on an Indian "sword":
which appears to be made to imitate a sawfish bill (and might not be made as a fighting weapon).
Short swords and daggers have decorative piercings much more often than swords. This might be because short weapons like these are usually not subjected to forces and stresses as great as those experienced by swords in use, and have much more strength than required, and therefore are less affected by loss of strength due to piercings:
https://i.pinimg.com/564x/26/05/74/260574549e969ec39a3ad47a6ad2c858.jpg
http://oriental-arms.co.il/item.php?id=8234 (with a copper blade, and thus more of a prestige/wealth/magic weapon than a fighting weapon)
https://i.pinimg.com/564x/9e/87/94/9e87949e1d9b639fe11b27b53ecc9794.jpg
In summary, axes, often with thicker blades than swords, give plenty of freedom for decorative cut-outs and piercings. Even without decorative work like this, axes often have a lot of material removed behind the blade (e.g., bearded axes) in order to reduce the weight of the head. Various axe-like polearms also offer a large and thick surface for such decoration. Extreme piercings and cut-outs are uncommon on swords, but are much more common on daggers - as noted above, swords are more likely to be pushing the limits of the material, and coulod be easily weakened too much by excessive piercing and cut-outs.
Where swords are pierced, it is often simpler than fantasy examples. Real-world axes and daggers can exceed the typical fantasy example in complexity, but tend to follow a quite different aesthetic to typical fantasy examples.