If your side gig is not part of your wife's business, you should not mix them. You certainly should not use her EIN for your side gig. You can get your own EIN so that you don't have to give out your Social Security number.
If you have separate business activities you should keep them separate. If you created a multimember LLC and ran all of your businesses under one LLC, then you must file one tax return for all of your combined businesses. That means that if one business has a loss and the other a profit, the profit will offset the loss and you won’t be able to deduct the loss on your personal return. Also, if one business is sued, all of your assets for all the businesses under the same LLC are vulnerable. Additionally, unless you live in a community property state, a multi member LLC has to file a form 1065 partnership return before you can file your personal 1040s and that requires a separate TurboTax program and an increased cost. Forming an LLC is actually a much more complex decision than you probably realized and you should consult an expert in your state before doing so.
As noted, if you are trying to report your own separate self-employment income that is a different type of business than your wife’s, you can use your personal Social Security number or you can get your own EIN online from the IRS in about 15 minutes.