Don't just assume they're identical without checking your specific accounts - rates do shift, and some money market accounts still edge out basic savings depending on your bank. That said, you're onto something real here. The gap between money market and regular savings has narrowed a lot since rates got competitive again. Most banks are offering pretty similar APYs now, so unless your money market account is genuinely paying more, the main "advantage" is mainly the slightly easier access to your cash if you need it (some money market accounts let you write checks or use a debit card, whereas savings accounts are more limited). But that's increasingly a marginal benefit.
The real question is whether your money market account has annoying fees or minimum balance requirements that a basic savings account doesn't. If you're paying monthly fees or have to keep a few grand sitting there to avoid penalties, you're definitely losing value. Consolidating into a high-yield savings account elsewhere might actually make more sense - shop around because rates vary wildly between banks, and some online banks are offering much better returns than traditional banks right now.
If you've been with the same bank since 2008, there's a decent chance you're not getting their current best rate anyway. Banks often give the best rates to new customers, not loyal ones. Take twenty minutes to compare what your current bank offers versus a couple competitors, then decide if keeping both accounts is worth the mental clutter or if one account somewhere actually makes more sense for your money.