Slugg your way through
"We were both slugging our ways through."
Can you please tell me what does it mean and what is the equivalent to it?

Thank you for help :)

Slugging is a way of saying fighting (in American English, to punch someone is to "slug" them). If that was the meaning, they're saying that you both had to fight (work) very hard, for the duration of the event or time period. A constant challenge.

It could also be that they meant "slog", which is more of a British term for a long, difficult walk. A similar meaning, but this one would suggest more of a boring, exhausting and unending task.

I think you mean "slog".  "Slog" means "walk slowly when you are tired".
"The soldiers slogged through the mud and rain, wishing they were home."

It is often used as a metaphor.
"My teacher assigned 1000 pages this week.  While all of my friends are having fun, I'll have to slog (my way) through 1000 pages of boring fiction."