What is the difference between はじめまして vs 始めよう?

Question

I was watching a Japanese TV show, and in the start of the episode, introductions were made between two people; then later in the show, they were playing a video game. In both instances, the phrase "Hajimemashite" was used.

Is it correct to use it in the second instance? Shouldn't they have said, "Hajimemashou" instead?

Research

How do you do?; I am glad to meet you​ - はじめまして on Jisho

From my understanding, this is used when meeting someone for the first time.

to start; to begin; to commence; to initiate; to originate - 始めよう on Jisho

Whereas, this one means to begin, e.g., to start a race, game, etc.

Answer 1

As the dictionary says, the two phrases are completely different, and they are never interchangeable.

• 初めまして is used when you meet someone for the first time. "Nice to meet you; Glad to meet you; How do you do"
• 始めましょう is just "Let's start (the show, lesson, etc)".

It's difficult to guess what was actually happening in your video. Maybe you misheard something (始まりました "(It) started" may sound similar if said quickly), or maybe they were just jokingly saying "hello" to a character in the game.