In Sweden we have something called "borgerlig vigsel"~"civil wedding" in the city hall. It existed long before gay marriage, which became legal just a few years ago.
We have the "burgerlijk huwelijk", "civil wedding". Open for gay and straight relationships.
And we have "geregistreerd partnerschap", "Civil Partnership". Open for gay and straight partners.
And, if you want to do it different, you can also go to a Notary public to arrange the legal matters of living with a partner, regardless of genders.