How with 4 items that gives more HP than hers, she is tankier and deals more damage?
The only way that Akali is really tanky is due to her base magic resistance, which is higher than average by 5 points. It's more that Galio is not very tanky for the class of champion he is; His base health is 62 higher and he gains 7 more hp per level. He also has 1 more armor point than normal. However, this is compared to Akali. Compared to a pure tank support, like Leona, his base armor is 23 points lower, and his MR growth is slightly lower.
Akali also built two of the highest health AP items in the game which do not bring any resistances. This means that while your health pools are similar, you have far more effective health than she does.
She deals more damage simply because the gold that you itemised toward armor resistance, she itemised toward AP. Mejais is also one of the most slot and gold efficient items in the game because it can give so much ability power, so if she had any stacks, that would also be why she was doing so much damage, especially when you consider that you were relying on your base magic resistance, which is quite low.
Let's say you're both level 18 for arguments sake, and you're both running the same items as you mentioned. You'll be taking about 35% reduced damage from magic damage, making your effective health round(((2466 base + 1100 from items) * 1.15) * 1.35) = 5536
. The additional 1.15 term is because with your W passive up you have an increased 15% effective health pool against magic damage.
Akali has more base magic resistance than Galio, but lower health scaling. Applying the same formula, Akali will have round((2285 base + 850 from items) * 1.37) = 4295
. This means that at maximum level, with the items you've listed, Galio can sustain around 1300 more damage than Akali can from magic damage.
Now, if we were to adjust this so that you were around 2 levels lower than Akali, which is likely for a support, then you'd be level 16, and the result would instead be round(((2183 + 1100) * 1.15) * 1.35) = 5096
, a reduction of 500 health, putting you much closer to Akali's level of tankiness.
This is the most likely reason why you felt so squishy against her. We can agree or disagree on whether Akali's damage is strong, but it's more that Galio without magic resistance is not very tanky, rather than Akali with health being very tanky.