Bidding wars, or "multiple bid situations," are common in NYC but have become the norm in many segments of the market as inventory lags behind buyer demand. While you're less likely to find yourself in a bidding war in a co-op than a condo, and more likely on a "mint" apartment than something that needs significant work, it's really case by case.
And it's also case by case how these are handled, and which bid the owners pick. The classic system is calling for "best and highest" bids by a certain date/time, and then one bid is chosen, generally with a couple others held as "backup." We don't have escalation clauses in NYC, so it's really a personal decision to choose the number where if you get the apartment you're elated, and if you lose it because someone bid $25k more you're ok because that was just beyond what you'd be willing to spend.
In condos the winning bid does tend to be the highest number, while co-ops have to consider a lot more factors (including debt to income, post-closing) in order to pass the board. But even in condos it's not JUST the highest number; it's the highest number without risky contingencies (risky for the seller, not the buyer) that also seems able to close. Because a number without the ability to deliver is just a number. I'll refer you to when a 10 year old child was negotiating with me on the back porch of the standalone house I sold. And another agent texted me "we'll do full ask" with no pre-approval, proof of funds, or terms. Obviously neither was an offer I could accept. Is Million Dollar Listing teaching people this is how real estate works? It isn't how real estate works.
In 99.9% of cases it makes sense to work with your own buyers agent, and in these situations it becomes increasingly important. I have been on the receiving end of many offers, and it is eye-opening how much better crafted and explained some are than others. The worst thing you can do is work with a bad agent. The next worst thing is to go into a multiple bid situation solo. And the best? Working (almost always for free) with an agent who knows wtf they're doing and makes it clear to the seller/seller's team that you will actually be able to make this purchase happen.
Are there bidding wars on rentals, too?
Unfortunately, the answer is yes. And much like with sales, it is due to both an overall shortage of housing and people with money not wanting to compromise on amenities, neighborhoods, or space/light. So you have 10 people fighting over every renovated apartment in the West Village, every cute Parlor floor unit in Fort Greene, and then an apartment on UES with a shower in the kitchen that's been on the market for 100 days below $2,000 a month.
At the end of the day, people with "fuck you money" can come in and overbid $500/mo to live where they want. If you aren't in that position and you keep getting outbid, you can keep pounding the pavement (if you have a flexible move date or plenty of time) or you can look elsewhere/give up some of the amenities you want. The "best" properties will always be competitive, and I saw bidding wars back in 2019 as well. The exception was May to October of 2020. For a brief, beautiful moment, tenants had all the power. And naturally I chose that time to run 300 rental units for a landlord.