Yes. Buy the cheap stuff. it is all the same except the label.
I assume your bike is made of either metal or carbon fiber? Neither will be eroded by any cleaning product on the market .... the "bike-specific" products are for people ... well, people other than me, let's be charitable. No reason not to spend $30 bucks on the exact same product another guy buys for $5, right?
What needs cleaning? Oil and grease ... same as all oil and grease. Anything which dissolves it is fine. Dirt and mud --- which comes off with water. What else.
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Lazyass is right on with the brushes, too.
I confess ... I once used laundry detergent when I was washing my bike next to the washer/dryer and was too lazy to walk across the garage to get the Simple Green. My bike survived.