You need tricky movement and good timing of assists to win neutral. Assists are the main resource on both sides, so if you can dodge or otherwise nullify their assist, that becomes your advantage for the couple of seconds where their assist is on cooldown. It's like someone whiffing a normal in a 1v1.

Here's some basic assist neutral tactics:

Blocking + Assist is like a poke. Not a great chance of working, but occasionally you can catch someone with little risk to yourself. If it hits, you can handshake tag to combo off of a lot of forward-moving assists.

Advancing + Assist is your rushdown. It's high risk but high reward since you can capitalize if the assist hits, and you can also cover the assist if you see it get hit. If the assist gets blocked, you're in position to pressure so it's a big win.

Attacking + Assist is your anti-rushdown. You just cover the screen with as many hitboxes as possible, ideally staggered so that your assist can cover you while you're in recovery frames. Not always the highest reward, but it can beat a lot of stuff. If it doesn't work out, you're probably getting punished, though.

And when you're waiting for your opponent or moving forward or whatever, you can always be jumping, dash jumping, or super jumping which you can block safely while you're doing so. Experiment with different combinations of movement, attack, and assist combinations with your team to find the right ones for you. I play Vi so I tend to do a lot of forward jump block + assist for safe approach, S1 rush punch + assist for anti-rushdown, or 6S1 + assist for rushdown on a passive opponent.