Patient Gamers

The ovious one: Tomb Raider

Since no one's mentioned it: Heavy Metal: F.A.K.K. 2

The older Resident Evil games (1, 2, Code Veronica, RE:Zero) have playable female protagonists. Actually most of the RE games have at least one segment where you play as a female character.

The swapping of two protagonists is actually something I wish they would bring back to the modern games in the franchise.

Phantasy Star. Love that game.

I put together a list of games for every letter of the alphabet with playable female characters that I've personally played and a lot of them are pretty old.

https://eldritch.cafe/@Gwendolyn/109348711301572294

Mischief Makers

The Siberia series, King's Quest IV, Parasite Eve series, Excalibur 2555 AD, Fear Effect series

The main protagonist in Final Fantasy 6 is female. There's a large number of games that age or newer with Female protags... older is a bit harder (but not too hard. You just unfortunately end up with a lot of fringe games or exploitive games)

And below, some cheatiness (a fairly exhaustive list of both :) )

1980s

1990s

Trails in the Sky FC and SC.

Also, FemShep was a popular option in Mass Effect for good reason.

At risk of posting twice but, I had to add this one as well:
Kyrandia 2: The Hand of Fate
A point-and-click adventure game. It was funny, colorful, and Zanthia is still awesome :) Made by Westwood of all people. It makes some references to the first, but you could play it on its own.

Eternal Darkness: Sanity's Requiem.

How about Mrs Mopp?

Metroid of course!

Also Ion Fury isn't old, but its made to look and feel old, and the protag is kick-ass

I haven't seen it mentioned so i'll go with Ion Fury. It's not really that old, but it's made on Build engine. The same was used in Duke Nukem 3D or Blood. Same roots means main protagonist have badass comments on lot of game occasions.

Siberia!!! And Dreamfall!!! Best games ever!!!

edit: also I saw Life is strange in the comments, this is a really good game. And also I remembered bayonetta!!!

Donkey Kong Country 3 if you count anthro characters. Dixie is also in DKC2 but there she is the sidekick and not the main character, not that there's much difference from a gameplay perspective.

Primal on PS2

Don't know if PS1 counts as old but Valkyrie profile is one of my favorite games and you play as a Valkyrie recruiting fallen warriors and training them up for Valhalla.
Star Ocean 2 (also on PS1) is another one of my favorites and you can choose between a guy or girl at the beginning, which changes the story a bit.

A lot of point-and-click adventure games do feature female protaganists. One that comes to mind that's not bee mentioned so far is Kathy Rain.

How in the world did y'all miss Silent Hill 3? It's October, and there are even other horror games mentioned. No excuse! Tsk tsk.

Napple Tale: Arisa in Daydream
A game for the Dreamcast that had a primarily female dev team. It was exclusive to Japan, but a translation patch was released a couple years ago.

Atelier series
There are a bunch of these. They’re RPGs based heavily around crafting. Actually, they basically pioneered crafting in RPGs.

Um Jammer Lammy
A spin-off of Parappa the Rapper. Lammy has only 15 minutes to make it to her band’s performance, but getting there won’t be easy.
Note that Stage 6 is censored in the North American release. Also as a bonus, completing the game unlocks Parappa mode, with new versions of every song after Stage 1.

Athena
Stick with the original arcade version. The NES port is infamously bad.

Psycho Soldier
A spin-off of Athena (sort of) starring Athena’s descendant/reincarnation (it depends on the writer), Athena Asamiya (and Sie Kensou as player 2’s character). Athena (and Kensou) would later become staples of the King of Fighters series. The game is notable for being the first game with a vocal soundtrack.

Baraduke
An arcade title from Namco. Being on this list is actually a pretty big spoiler because the fact that the protagonist is a woman isn’t revealed until the end of the game. Now that probably sounds like a ripoff of Metroid, but here’s the thing: Baraduke did it first.

Koudelka
An obscure horror RPG for the PlayStation.
The game is an unhappy compromise between the creator and the dev team. One side wanting to make a survival horror game and the other going for something like Final Fantasy VII. I’ve never actually gotten very far in this one because I hate the battle system. Also, this game has connections to the later Shadow Hearts series.

Literally every game in the Otome genre
Probably not what you’re looking for, but technically fits your criteria of having a female protagonist.

I might add more to this post if I can think of something else.

If you accept games where you fully create the character, you've got a ton of RPGs. Baldur's gate, elder scrolls, arcanum, fallout.