Pretty much just that. I never knew why. Some people have said "because of the two-party system" but I don't really know what it means by the system, because there are definitely more than two parties, and people know of them. That doesn't really seem to excuse not identifying as a Social Democrat.
Why Bernie identifies as a Democrat is due to the structural dominance of the two major parties in US politics. There have been individuals that have run for office as candidates of other parties (such as the Liberatrian Party and the Green Party), but most observers don't give them much chance at being elected to the Office of the President. Politicians from third parties have had some success at being elected to Congress, but notably not so much in the modern era. Because of this, Bernie most likely came to the conclusion that if he wants a chance to actually win, he should affiliate with one of the two major parties. Between them the Democrats align more closely to his own views and positions.
The rest of your question isn't quite right, not everyone calls him a socialist. Those who do are generally trying to define him as one in the hopes that those who are afraid of that label would react negatively. You are right that he considers himself to be a "democratic socialist", but their organization and membership are low and they are not well known outside of their own circles, so running for President underneath their banner probably would not help him actually win.
Bernie Sanders is actually just loosely associated with the Democratic Party. He officially holds his seat as a Senator as an independent.
However, during the current and previous Presidential pre-elections, he sought the nomination of the Democratic party. He can do that without actually being a member of the party. Everyone who is eligible to be elected President can apply.
The reason why he choose to do that with the Democrats and not as an independent candidate or as a candidate of a minor party which more aligns with his political ideas is because there was no Presidential election in recent history where a candidate who wasn't nominated by the two main parties got even close to winning just a single state. A presidential candidate who isn't nominated by either the Republican Party or the Democratic Party has no realistic chance to get elected. So if he seriously wants to get elected President, he needs to get the nomination of either the Republican party or the Democratic party. It appears that of those two options, the Democrats are still his preferred choice.
And then there is another problem: vote splitting.
If he would actually run for a different party and try to turn the two-party race for Presidency into a three-party race, he would likely hurt his political ideas because he would split the progressive vote.
Sanders is so far away from conservativism, that he has no chance to realistically steal votes from the Republican candidate. The voters he could realistically convince to vote for him are those people who would otherwise vote whoever the Democratic Party nominates. So there is a good chance that Sanders and the Democratic candidate have more electoral votes than the Republican candidate combined, but less votes individually. So the Republican candidate would win, even though the majority of the country would prefer someone else. Sanders likely believes that if he can not be President, he would rather prefer a Democrat in the white house than a Republican.
He's asking registered Democrats to vote for him in the Democrat Primary(ies), with the view to the broad Democrat party apparatus backing him in the election.
I.e. he wants to be the Democrat candidate, not the Social Democrat candidate, not the Socialist candidate, not an Independent candidate.