What is the function of 「だと」in this sentence?
The だと consists of the copula 「だ」 (or, the assertive auxiliary / 断定の助動詞「だ」) and
the conjunctive particle / 接続助詞「と」, meaning "if" or "when".
So これだと literally means "If/When (it) is this", ⇒ "If/When you choose/take/use this", or "With this".
You could rephrase your sentence as これなら乗り換えなし. "If it's this, ⇒ With this, you don't have to change trains."
Sometimes I hear 「だって」 in a rather forceful/insistent tone. Are they the same?
No, the だと here is quite different from だって.
The だと that can be rephrased as だって, as in 「なんだって？」≂「なんだと？」, is the copula 「だ」 + the quotative particle / 引用の格助詞「と」.
これだって can mean "even if (it) is this" / "even this" (≂ これでも), or "this one, too" (≂ これも). The だって here is made of the copula 「だ」 + the binding particle / 係助詞「とて」.