毎(まい)～の様(よう)にAlmost every ~, Nearly every ~, On a ~ basis

These expressions are related through the ように in まい～のように, which is the same as in ように・ような, both expressing comparison. まい～のように is used with 'every~' time expressions to mean 'almost every (A)', e.g. 毎日のように 'almost every day'. In contrast, ように・ような is used for general comparisons, e.g. 春のように暖かい ('It's warm like spring.').

Both phrases express frequency but differ in usage. ごとに attaches to time expressions, nouns, or verbs to mean 'every (A),' e.g. ３時間ごとに 'every three hours' or 失敗するごとに 'every time (you) fail'. まい～のように is used only with time words that take 毎 and adds emphasis, meaning 'almost every (A)', e.g. 毎日のように 'nearly every day'.

Both まい～のように and たびに express frequency and the idea of 'every,' but with different nuances. まい～のように is used with 'every~' time expressions to mean 'almost every (A)', e.g. 毎日のように 'almost every day'. In contrast, たびに follows verbs or noun phrases to mean 'every time (A)', e.g. 彼に会うたびに 'every time I see him' or 旅行のたびに 'every time I take a trip'.

Both まい～のように and おきに express frequency, but in different ways. まい～のように is used with 'every~' time expressions to mean 'almost every (A)', e.g. 毎日のように 'almost every day'. In contrast, おきに attaches to a number + counter expression to mean 'at intervals of (A),' e.g. 一日おきに 'every other day' or 'at intervals of one day'.

まい～のように and につき both express frequency, but in different ways. まい～のように is used with 'every~' time expressions to mean 'almost every (A)', e.g. 毎日のように 'almost every day'. In contrast, につき follows time/quantity expressions to mean 'each' or 'every,' e.g. 30分につき 'for every 30 minutes'. につき can also mean 'due to', which is unrelated to frequency.