Is saying "I will recite Surat Al-Baqarah my entire life" considered a vow to Allah, and does the Hadith "Whoever vows to obey Allah, let him obey Him" apply to it? Is it a condition for a vow to be in an explicit form, such as "I vow"?

The vow requires a formula that signifies commitment, such as: "It is a duty upon me for Allah" or "I owe such-and-such a vow." It is not constituted by mere intention unless it is a metaphor accompanied by an intention. However, the phrase "I will do such-and-such" is not considered a vow unless the intention behind it is to make a vow. A vow is constituted by uttering its لفظ (wording), even if there is no accompanying intention.