I'd expect another inquiry. Aside from that, IMO, it doesn't hurt to join more than one. Just keep an eye on any fees.
I just joined my 2nd credit union. The 1st Credit Union I have been a member of for about 5 years. They are very conservative with lending, etc. So I decided to join another as more of an experiment. I joined and applied for a personal loan on the same day last week, they approved me. Then today, they approved me for a credit card. So it depends on your individual needs, if you're looking for checking/savings, loans/credit, investments/insurance, etc. it may be beneficial to join two.
There can definitely be advantages to having two.
Credit unions that are more conservative in their lending practices may be able to pay higher dividends on savings, money market, etc.
Meanwhile, those same credit unions may be harder to get a loan from. Whereas a CU that pays out lower amounts may be more liberal in extending to you the credit you need.
Horses for courses, as the saying goes.
I learned from experience in the financial crisis that having everything in one or two financial institutions can hurt you when one decides to change course. There's also the risk of mergers etc. causing a negative turn.
I choose to belong to two CUs (NFCU and my job CU) and USAA (more like a CU than a bank).