Sign in to your account is correct. To sign in is a phrasal verb.
The phrasal verb “To sign in” can take different prepositions and to is just one example.
Sign in at the desk.
Sign in over there.
You need to sign in by 9 pm.
We do not combine in and at and neither do we combine in and to.
You need to sign in to your account
You need to sign into your account.
When to use into?
Into often has the meaning of transformation.
He turned into a monster.
He turned in to the gas station.
We also use into when we combine the prepositions in and to(but NOT when in is part of a phrasal verb).
Let’s move into the sitting room.
In this case, we need the preposition in to show the space of the sitting room and to to show movement. This is the situation when you can combine “in” and “to” to form into.
Sign-in as a noun
Phrasal verbs can also be used as nouns(especially in spoken English).
Where is the sign-in sheet?
What is your sign-in number?
In more informal English, it is common to drop the hyphen.
Signed into or Signed in to?
“Signed in to” is correct and should always be used.