Mi hermano y yo jugamos al mismo videojuego en la consola del salón.

With games and sports, Spanish normally uses jugar + a + [game/sport].

• jugar al fútbol – to play football
• jugar a las cartas – to play cards
• jugar al mismo videojuego – to play the same video game

a + el contracts to al, so:

• jugar a el mismo videojuego → jugar al mismo videojuego

You will sometimes hear jugar el videojuego in some varieties, but the standard, safest form to learn is jugar a [juego/deporte].

al is just the contraction of a + el:

• a = to/at (here it goes with jugar)
• el = the (masculine singular article)

So:

• jugamos al mismo videojuego = jugamos a el mismo videojuego
  (= “we play the same video game”)

Any time a comes right before el, in standard Spanish it becomes al:

• Voy al parque. – Voy a el parque.
• Llamo al médico. – Llamo a el médico.

Again, del is a contraction:

• de + el → del

So:

• la consola del salón = “the console of the living room”
  (= the console that is in / belongs to the living room)

You use del every time de comes directly before el:

• el color del coche – the colour of the car
• la puerta del jardín – the garden door

jugamos can be either:

• present, 1st person plural – (nosotros) jugamos = we play / we are playing
• preterite (simple past), 1st person plural – (nosotros) jugamos = we played

In your sentence, with no time expression, the natural reading is present:
Mi hermano y yo jugamos al mismo videojuego… = My brother and I play the same video game…

To make it clearly past, Spanish usually adds a time word or context:

• Ayer mi hermano y yo jugamos al mismo videojuego. – Yesterday my brother and I played…

Spanish normally omits subject pronouns because the verb ending already shows the subject:

• (Nosotros) jugamos – we play

You cannot say Nosotros mi hermano y yo jugamos…; that sounds wrong and redundant.

You can say:

• Mi hermano y yo jugamos… (most natural)
• Nosotros jugamos… (if you need to emphasize we)

But not both together.

Spanish prefers to mention other people before “yo” in a list, as a matter of politeness and style:

• Mi hermano y yo – natural
• Yo y mi hermano – grammatically possible but sounds a bit self-centred or childish

So you would normally say:

• Mis amigos y yo – my friends and I
• Mis padres y yo – my parents and I

When yo is part of the subject, Spanish uses 1st person plural:

• Mi hermano y yo jugamos… – my brother and I play…
• Tú y yo jugamos… – you and I play…
• Mis amigos y yo jugamos… – my friends and I play…

If the subject doesn’t include yo, you use 3rd person plural:

• Mi hermano y mi primo juegan… – my brother and my cousin play…
• Ellos juegan… – they play…

There are two different prepositions in play:

1. jugar a [juego/deporte] – play a game/sport

   • jugar al mismo videojuego

2. en [lugar/objeto] – on / in / at (location or medium)

   • en la consola – on the console

So you combine them:

• jugar al videojuego en la consola – to play the video game on the console

Using a la consola would suggest play the console as if the console itself were the “game”, which is not how it’s expressed.

mismo means “same” here:

• jugamos al videojuego – we play the video game
• jugamos al mismo videojuego – we play the same video game

Grammatically, mismo is an adjective and agrees in gender and number with the noun:

• el mismo videojuego – the same video game (masc. sing.)
• la misma consola – the same console (fem. sing.)
• los mismos videojuegos – the same video games (masc. pl.)
• las mismas consolas – the same consoles (fem. pl.)

This mismo is different from reflexive expressions like a sí mismo (himself).

In Spanish, videojuego is written as one word:

• video
  • juego → videojuego

Spanish often forms compound nouns this way:

• paraguas – para + aguas (umbrella)
• sacapuntas – saca + puntas (pencil sharpener)

You may hear juego de video in some Latin American usage, but videojuego is the standard, especially in Spain.

In Peninsular Spanish, el salón in a home context usually means the living room / lounge.

So la consola del salón is the (game) console in the living room.

Other uses:

• el salón de actos – auditorium, assembly hall
• salón de belleza – beauty salon (but here salón needs another word)

For a hair salon specifically, you’d say peluquería, not just salón.

Both are possible, but they focus on slightly different things:

• la consola del salón
  = literally “the console of the living room”
  → it sounds like the console that belongs to / is kept in that room.

• la consola en el salón
  = “the console in the living room”
  → just states the location, without that sense of belonging.

In everyday speech about things normally kept in a room (the TV, the sofa, the console), del salón is very natural in Spain.