Your account is the bidder, not the device. Open the same show on a phone, a laptop and a tablet and you are still one bidder with one position in the auction.
What having both open does
Nothing to your bids. Bids live on the server. Both screens are windows onto the same auction, and both show the same current price, the same clock and the same standing.
One of them may lag. Whichever tab has been in the background longer is more likely to be showing a stale price or a paused video. Phone browsers throttle background tabs aggressively.
That is the actual risk here, and it is not about double-bidding: it is about bidding from the screen that is behind. A price you can see on a backgrounded laptop may already be out of date.
The rule to follow
Bid from the screen you are actually looking at, and refresh it if you have just come back to it.
If you want both open — laptop for the video, phone for the bidding — that is a reasonable setup. Just do all the bidding on one of them.
What you cannot do
Bid against yourself to push a price up, through two devices or two accounts. One account is one bidder, and a second account would not help: you cannot bid on your own item, and nor can anyone on your store's team. Can I have two accounts?
Share an account with somebody who also wants to bid. Two people on one login is one bidder, and both of you are bound by whatever either of you bids. If a friend wants to bid, they need their own account — that is also the only way they get their own notifications and orders.
Signing out everywhere
If you are signed in somewhere you no longer trust, Signed-in devices on the security screen shows what is active, and Sign out all ends every session on every device — Signing out everywhere ends every session, on every device.
A bid already placed is not undone by signing out. It stands, because it was yours when you made it.
Leaving one open after a win
A payment window runs from the win, not from whichever device you happen to be holding. So if you win on the laptop and walk away with your phone, the clock is still running.
Check your orders on whatever device you have. How will I know if I won?
More: if I leave the show, do I lose my bid, how bidding works, can I buy and sell with one account.