I have the same issue. This problem is currently my major reason for the fact that I am likely going to switch both my personal and company accounts to a competitor which handles these situations more appropriately. I am not the only person within our company who has this issue. While this can be solved on the desktop using profiles in Chrome, the fact that I cannot sign into multiple accounts on iOS or Android limits the efficacy of LastPass as a solution for myself and my customers.
Please note that I've been a paying LastPass Premium customer since 2011, a free tier customer for 2 years prior to that, have earned quite a bit of referral bonuses, and made the decision for our company to become a LastPass Enterprise customer for the past 2 years. I was a loyal customer, but support requests pointing out this issue are met with responses that indicated that there was something wrong on my device's end. Clearly, according to @AyaLP, this is not the case.
Please fix this issue.