Saving money together is a nice symbolic gesture and a sign that you see some form of future together. Maybe you’ve just moved in together and need to save for a new couch or bed. It’s a given that your shared stuff should be bought from a together-account.
For those of you that have been together for a while, you probably have bigger goals. Maybe the next step is a car, an apartment or maybe even savings for your children. Sharing your savings can strengthen your bond and at the same time give you an insight into eachothers economy.
Everyone should have a little money saved for themselves regardless of relationship status. A buffer in case anything unpredicted would happen. Your own savings don't have to be secretive, but it is necessary for your own comfort. Your own savings could also be used for the more expensive stuff you want for yourself, so you can reward yourself without it affecting the household budget.
Make sure that you both know the conditions when saving together. A regular shared account may give you both access to the money and an overview of the account. There may be various conditions depending on where you open an account. A big question is: what happens if you break up.
ISK-accounts for stocks are registered in one persons name. If you choose to save in stocks and mutual funds you’ll have to write an agreement to make sure the money gets evenly split.